Ch.1: Math fundamentals
Chapter 1 of the book "No Bullshit Guide to Linear Algebra" by Ivan Savov Solving equations Numbers Variables Functions and their inverses Basic rules of algebra Solving quadratic equations The Cartesian plane Functions Function reference Polynomials Trigonometry Trigonometric identities Geometry Circle Solving systems of linear equations Set notation
234 flashcards · Mathematics · by lakesare
Part of Textbooks › No Bullshit Guide to Linear Algebra (Ivan Savov) · 10 courses
[Math fundamentals] In what sense is linear algebra an extension of high-school maths with numbers and functions?
Linear algebra does the same kind of maths with vectors and linear transformations in place of numbers and functions. A function takes numbers as inputs and produces numbers as outputs; a linear transformation takes vectors as inputs and produces vectors as outputs.
[Solving equations] What does it mean to solve an equation for xxx ?
To transform the equation into the form x=(only numbers)x = \text{(only numbers)}x=(only numbers) , through a sequence of equations that are each true when the original one is. Example: x2−4=45x^2 - 4 = 45x2−4=45 is solved once it is written as x=7x = 7x=7 or x=−7x = -7x=−7 .
[Solving equations] Which rule keeps an equation true while you manipulate it?
Every change applied to the left side must also be applied to the right side. The sign === states that everything on the left equals everything on the right; changing only one side breaks that.Example: from x2−4=45x^2 - 4 = 45x2−4=45 , adding 4 to both sides gives x2=49x^2 = 49x2=49 .
[Solving equations] How do you remove an operation that stands in the way of isolating xxx ?
Apply the inverse operation to both sides. Example: subtracting 4 is undone by adding 4; squaring is undone by taking the square root.
[Solving equations] What is the absolute value ∣x∣|x|∣x∣ of a number xxx ?
xxx without its sign: ∣x∣=x|x| = x∣x∣=x if x≥0x \ge 0x≥0 , and ∣x∣=−x|x| = -x∣x∣=−x if x<0x < 0x<0 . Example: ∣5∣=5|5| = 5∣5∣=5 and ∣−5∣=5|-5| = 5∣−5∣=5 .
[Solving equations] Simplify x2\sqrt{x^2}x2 .
x2=∣x∣\boldsymbol{\sqrt{x^2} = |x|}x2=∣x∣ xxx and −x-x−x have the same square, and x\sqrt{\phantom{x}}x returns the nonnegative one.Example: (−7)2=49=7=∣−7∣\sqrt{(-7)^2} = \sqrt{49} = 7 = |-7|(−7)2=49=7=∣−7∣ .
[Solving equations] Solve x2−4=45x^2 - 4 = 45x2−4=45 .
x=7\boldsymbol{x = 7}x=7 or x=−7\boldsymbol{x = -7}x=−7 Add 4: x2=49x^2 = 49x2=49 . Take the square root of both sides: ∣x∣=7|x| = 7∣x∣=7 .Both are solutions because 72=(−7)2=497^2 = (-7)^2 = 4972=(−7)2=49 .
[Numbers] What are the five standard sets of numbers, from smallest to largest?
Natural numbers N\mathbb{N}N N={0,1,2,3,… }\mathbb{N} = \{0, 1, 2, 3, \dots\}N={0,1,2,3,…} (this book includes 0) Integers Z\mathbb{Z}Z Z={…,−2,−1,0,1,2,… }\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}Z={…,−2,−1,0,1,2,…} Rational numbers Q\mathbb{Q}Q Fractions of integers. Example: 53, 227, 1.5, 0.125, −7\frac{5}{3}, \ \frac{22}{7}, \ 1.5, \ 0.125, \ -735, 722, 1.5, 0.125, −7 Real numbers R\mathbb{R}R The rationals together with the irrational numbers. Example: −1, 0, 2, e, π, 4.94…-1, \ 0,…
[Numbers] What makes each of N\mathbb{N}N , Z\mathbb{Z}Z , Q\mathbb{Q}Q , R\mathbb{R}R insufficient, so that the next larger set of numbers is needed?
N\mathbb{N}N to Z\mathbb{Z}Z : subtraction Example: 5−7=−2∉N5 - 7 = -2 \notin \mathbb{N}5−7=−2∈/N Z\mathbb{Z}Z to Q\mathbb{Q}Q : division Example: 1÷4=14∉Z1 \div 4 = \frac{1}{4} \notin \mathbb{Z}1÷4=41∈/Z Q\mathbb{Q}Q to R\mathbb{R}R : irrational quantities Example: 2\sqrt{2}2 , the diagonal of a square with side 1 R\mathbb{R}R to C\mathbb{C}C : square root of a negative number Example: −1∉R\sqrt{-1} \notin \mathbb{R}−1∈/R N\mathbb{N}N suffices for counting and is closed under addition; Z\ma…
[Numbers] Write the set of rational numbers Q\mathbb{Q}Q in set notation.
Q={z∣z=xy, x,y∈Z, y≠0}\boldsymbol{\mathbb{Q} = \{ z \mid z = \frac{x}{y}, \ x, y \in \mathbb{Z}, \ y \ne 0 \}}Q={z∣z=yx, x,y∈Z, y=0} Example: 0.125=180.125 = \frac{1}{8}0.125=81 and −7=−71-7 = \frac{-7}{1}−7=1−7 are rational.
[Numbers] What is an irrational number, and what does its decimal expansion look like?
A real number that is not a fraction xy\frac{x}{y}yx of integers. Its decimal expansion is infinite and does not repeat. Example: 2=1.4142135…\sqrt{2} = 1.4142135\dots2=1.4142135… and π=3.14159265…\pi = 3.14159265\dotsπ=3.14159265… (the ratio of a circle's circumference to its diameter).
[Numbers] What do the commutative and associative laws of addition say?
Commutative: a+b=b+aa + b = b + aa+b=b+a . Associative: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)(a+b)+c=a+(b+c) . So a long sum a+b+ca + b + ca+b+c can be computed in any order. Multiplication satisfies both laws too: ab=baab = baab=ba , (ab)c=a(bc)(ab)c = a(bc)(ab)c=a(bc) .Example: (2+5)+3=2+(5+3)=10(2 + 5) + 3 = 2 + (5 + 3) = 10(2+5)+3=2+(5+3)=10 .
[Numbers] How is the product ababab of natural numbers defined through addition?
ab=a+a+⋯+aundefinedb times=b+b+⋯+bundefineda times\boldsymbol{ab = \underbrace{a + a + \cdots + a}_{b \text{ times}} = \underbrace{b + b + \cdots + b}_{a \text{ times}}}ab=b timesa+a+⋯+a=a timesb+b+⋯+b Example: 3⋅4=3+3+3+3=4+4+4=123 \cdot 4 = 3 + 3 + 3 + 3 = 4 + 4 + 4 = 123⋅4=3+3+3+3=4+4+4=12 .
[Numbers] What does the product ababab mean geometrically, and why is aa=a2aa = a^2aa=a2 read " aaa squared"?
ababab is the area of a rectangle with base aaa and height bbb ; with height equal to base the rectangle is a square of area a2a^2a2 . Example: a 3×33 \times 33×3 square has area 32=93^2 = 932=9 .
[Numbers] What is a/b=aba / b = \frac{a}{b}a/b=ba , and why is division by 0 undefined?
One bbb -th of aaa : divide aaa into bbb equal parts and take one part. There is no way to divide something into zero equal parts. Division is the inverse operation of multiplication.Example: 12/3=412 / 3 = 412/3=4 , since 3⋅4=123 \cdot 4 = 123⋅4=12 .
[Numbers] What do aba^bab and a−ba^{-b}a−b mean (for a natural number bbb )?
ab=aa⋯aundefinedb times\boldsymbol{a^b = \underbrace{a a \cdots a}_{b \text{ times}}}ab=b timesaa⋯a and a−b=1ab\boldsymbol{a^{-b} = \frac{1}{a^b}}a−b=ab1 The minus sign in an exponent means "divide by", not "subtract".Example: 23=82^3 = 823=8 and 2−3=182^{-3} = \frac{1}{8}2−3=81 .
[Numbers] What does the fractional exponent a1na^{\frac{1}{n}}an1 mean?
The nnn -th root of aaa : a1n≡an\boldsymbol{a^{\frac{1}{n}} \equiv \sqrt[n]{a}}an1≡na , the number which multiplied together nnn times gives aaa . Example: 813=83=28^{\frac{1}{3}} = \sqrt[3]{8} = 2831=38=2 , since 2⋅2⋅2=82 \cdot 2 \cdot 2 = 82⋅2⋅2=8 .Example: a14=a4=aa^{\frac{1}{4}} = \sqrt[4]{a} = \sqrt{\sqrt{a}}a41=4a=a , since 14=12⋅12\frac{1}{4} = \frac{1}{2} \cdot \frac{1}{2}41=21⋅21 .
[Numbers] Why is xn\sqrt[n]{x}nx the inverse function of xnx^nxn ?
Multiplying a1na^{\frac{1}{n}}an1 by itself nnn times gives ann=a1=aa^{\frac{n}{n}} = a^1 = aann=a1=a , so (an)n=a\left(\sqrt[n]{a}\right)^n = a(na)n=a . Example: (53)3=513+13+13=5\left(\sqrt[3]{5}\right)^3 = 5^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = 5(35)3=531+31+31=5 .
[Numbers] What does amna^{\frac{m}{n}}anm equal in terms of roots?
amn=(an)m=amn\boldsymbol{a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}}anm=(na)m=nam mn=m⋅1n=1n⋅m\frac{m}{n} = m \cdot \frac{1}{n} = \frac{1}{n} \cdot mnm=m⋅n1=n1⋅m , and the order of the two operations does not matter.Example: 823=(83)2=22=48^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4832=(38)2=22=4 , and 823=643=4\sqrt[3]{8^2} = \sqrt[3]{64} = 4382=364=4 .Example: a−12=1aa^{-\frac{1}{2}} = \frac{1}{\sqrt{a}}a−21=a1 , so 4−12=124^{-\frac{1}{2}} = \frac{1}{2…
[Numbers] What is the difference between writing an=a1n\sqrt[n]{a} = a^{\frac{1}{n}}na=an1 and an≡a1n\sqrt[n]{a} \equiv a^{\frac{1}{n}}na≡an1 ?
=== says two expressions have the same value; ≡\equiv≡ ("is equivalent to") says they are two notations for the same mathematical object. ≡\equiv≡ is the stronger relation.
[Numbers] In which order are the basic algebra operations performed (operator precedence)?
1. exponents and roots, 2. products and divisions, 3. additions and subtractions Parentheses override this order.Example: 5×32+13=5⋅9+13=585 \times 3^2 + 13 = 5 \cdot 9 + 13 = 585×32+13=5⋅9+13=58 , but (5×3)2+13=225+13=238(5 \times 3)^2 + 13 = 225 + 13 = 238(5×3)2+13=225+13=238 .
[Variables] What is a variable?
A placeholder name for any number or for an unknown. Variables let us compute with quantities whose value we don't know (abstraction).Example: a glass holds an unknown xxx ml; after nnn such glasses you drank nxnxnx ml.
[Variables] Which quantities do the variable names xxx , vvv , θ\thetaθ and φ\varphiφ , xix_ixi and xfx_fxf , XXX , and CCC , PPP , RRR usually denote?
xxx The unknown in an equation; also a function's input or an object's position. vvv Velocity (in physics). θ\thetaθ , φ\varphiφ Angles (Greek theta and phi). xix_ixi , xfx_fxf Initial and final position of an object (in physics). XXX A random variable (in probability theory). CCC , PPP , RRR Cost, profit and revenue (in business).
[Variables] What is variable substitution (change of variables)?
Replacing an expression in the unknown by a new variable, to get a simpler equation; solve for the new variable, then convert back. Example: in 65−x=x\frac{6}{5 - \sqrt{x}} = \sqrt{x}5−x6=x , let u=xu = \sqrt{x}u=x ; the equation becomes 65−u=u\frac{6}{5 - u} = u5−u6=u , with no square roots.
[Variables] Solve 65−x=x\dfrac{6}{5 - \sqrt{x}} = \sqrt{x}5−x6=x by substituting u=xu = \sqrt{x}u=x .
x=4\boldsymbol{x = 4}x=4 or x=9\boldsymbol{x = 9}x=9 65−u=u\frac{6}{5 - u} = u5−u6=u ; multiply by 5−u5 - u5−u : 6=5u−u26 = 5u - u^26=5u−u2 , so u2−5u+6=(u−2)(u−3)=0u^2 - 5u + 6 = (u - 2)(u - 3) = 0u2−5u+6=(u−2)(u−3)=0 . u=2u = 2u=2 or u=3u = 3u=3 , and x=u2x = u^2x=u2 .Check: 65−2=2=4\frac{6}{5 - 2} = 2 = \sqrt{4}5−26=2=4 and 65−3=3=9\frac{6}{5 - 3} = 3 = \sqrt{9}5−36=3=9 .
[Variables] In a long calculation with a mass of 140 kg, why write the variable mmm until the end instead of the number?
mmm is shorter to write in every step, and the value 140 kg is substituted only once, in the last step. This is compact notation: symbolic manipulation keeps the steps simple.
[Functions and their inverses] fff is a function and ccc a constant. How do you solve f(x)=cf(x) = cf(x)=c for xxx ?
Apply the inverse function f−1f^{-1}f−1 to both sides: f−1(f(x))=xf^{-1}(f(x)) = xf−1(f(x))=x , so x=f−1(c)\boldsymbol{x = f^{-1}(c)}x=f−1(c) . This requires f−1f^{-1}f−1 to be defined at the input ccc .Example: 2x=82^x = 82x=8 gives x=log28=3x = \log_2 8 = 3x=log28=3 .
[Functions and their inverses] What does f−1(y)f^{-1}(y)f−1(y) mean for a function fff ? Is it 1f(y)\frac{1}{f(y)}f(y)1 ?
f−1(y)f^{-1}(y)f−1(y) is the number xxx with f(x)=yf(x) = yf(x)=y . No, it is not 1f(y)=(f(y))−1\frac{1}{f(y)} = (f(y))^{-1}f(y)1=(f(y))−1 . The notation is borrowed from numbers: a−1a^{-1}a−1 undoes multiplication by aaa , since a−1ax=xa^{-1} a x = xa−1ax=x .Example: for f(x)=2xf(x) = 2xf(x)=2x : f−1(6)=3f^{-1}(6) = 3f−1(6)=3 , while 1f(6)=112\frac{1}{f(6)} = \frac{1}{12}f(6)1=121 .
[Functions and their inverses] c>0c > 0c>0 . Why does x2=cx^2 = cx2=c have two solutions although the inverse of x2x^2x2 is x\sqrt{x}x ?
x2x^2x2 maps both xxx and −x-x−x to the same output, so x=cx = \sqrt{c}x=c and x=−cx = -\sqrt{c}x=−c both solve it: x=±c\boldsymbol{x = \pm\sqrt{c}}x=±c . Example: x2=9x^2 = 9x2=9 gives x=±3x = \pm 3x=±3 .
[Functions and their inverses] What are the inverses of x+2x + 2x+2 , 2x\ 2x 2x , −x\ -x −x and x2\ x^2 x2 ?
x−2\boldsymbol{x - 2}x−2 , 12x\ \boldsymbol{\frac{1}{2}x} 21x , −x\ \boldsymbol{-x} −x , ±x\ \boldsymbol{\pm\sqrt{x}} ±x −x-x−x is its own inverse: −(−x)=x-(-x) = x−(−x)=x .
[Functions and their inverses] What is the inverse of f(x)=3x+5f(x) = 3x + 5f(x)=3x+5 ?
f−1(x)=13(x−5)\boldsymbol{f^{-1}(x) = \frac{1}{3}(x - 5)}f−1(x)=31(x−5) fff multiplies by 3, then adds 5; the inverse undoes these in reverse order: subtract 5, then divide by 3.Example: f(2)=11f(2) = 11f(2)=11 and f−1(11)=13⋅6=2f^{-1}(11) = \frac{1}{3} \cdot 6 = 2f−1(11)=31⋅6=2 .
[Functions and their inverses] What are the inverses of 2x2^x2x , ax\ a^x ax and exe^xex ?
log2(x)\boldsymbol{\log_2(x)}log2(x) , loga(x)\ \boldsymbol{\log_a(x)} loga(x) , ln(x)≡loge(x)\ \boldsymbol{\ln(x) \equiv \log_e(x)} ln(x)≡loge(x) exe^xex is also written exp(x)\exp(x)exp(x) .Example: 25=322^5 = 3225=32 and log2(32)=5\log_2(32) = 5log2(32)=5 .
[Functions and their inverses] What are the inverses of sin(x)\sin(x)sin(x) and cos(x)\cos(x)cos(x) , and how are they written?
sin−1(x)≡arcsin(x)\boldsymbol{\sin^{-1}(x) \equiv \arcsin(x)}sin−1(x)≡arcsin(x) and cos−1(x)≡arccos(x)\boldsymbol{\cos^{-1}(x) \equiv \arccos(x)}cos−1(x)≡arccos(x) Example: sin(30∘)=12\sin(30^\circ) = \frac{1}{2}sin(30∘)=21 and arcsin(12)=30∘\arcsin\left(\frac{1}{2}\right) = 30^\circarcsin(21)=30∘ .
[Functions and their inverses] ggg is the inverse function of fff . What is the inverse function of ggg ?
f\boldsymbol{f}f The function-inverse relationship goes both ways.Example: loga(x)\log_a(x)loga(x) is the inverse of axa^xax , and axa^xax is the inverse of loga(x)\log_a(x)loga(x) .
[Functions and their inverses] What is the general procedure for isolating xxx when it sits inside several nested functions?
Undo the outermost function on the side of xxx by applying its inverse to both sides; repeat until xxx stands alone. Keep constants that don't involve xxx as symbols and compute numbers only in the last step.Example: 2x+1=3\sqrt{2x + 1} = 32x+1=3 : square, 2x+1=92x + 1 = 92x+1=9 ; subtract 1, 2x=82x = 82x=8 ; divide by 2, x=4x = 4x=4 .
[Functions and their inverses] Ψ\PsiΨ is some function. Solve for xxx : log5(3+6x−7)=34+sin(5.5)−Ψ(1)\log_5\left(3 + \sqrt{6\sqrt{x} - 7}\right) = 34 + \sin(5.5) - \Psi(1)log5(3+6x−7)=34+sin(5.5)−Ψ(1)
x=[16((534+sin(5.5)−Ψ(1)−3)2+7)]2\boldsymbol{x = \left[ \frac{1}{6} \left( \left( 5^{34 + \sin(5.5) - \Psi(1)} - 3 \right)^2 + 7 \right) \right]^2}x=[61((534+sin(5.5)−Ψ(1)−3)2+7)]2 Write K=34+sin(5.5)−Ψ(1)K = 34 + \sin(5.5) - \Psi(1)K=34+sin(5.5)−Ψ(1) ; the right side contains no xxx , so it stays as it is.Apply 5(⋅)5^{(\cdot)}5(⋅) : 3+6x−7=5K3 + \sqrt{6\sqrt{x} - 7} = 5^K3+6x−7=5K . Subtract 3, square: 6x−7=(5K−3)26\sqrt{x} - 7 = (5^K - 3)^26x−7=(5K−3)2 .Add 7, divide by 6, square: x=[16…
[Functions and their inverses] When does the method "apply inverse functions to both sides until xxx is alone" fail?
When xxx appears in several places in the equation. Then other techniques are needed, e.g. the quadratic formula for ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 ; for third-degree equations ax3+bx2+cx+d=0ax^3 + bx^2 + cx + d = 0ax3+bx2+cx+d=0 one usually uses a computer.Example: in x2=5x−6x^2 = 5x - 6x2=5x−6 no single inverse function isolates xxx .
[Basic rules of algebra] Things added together in an expression are called terms; if x=abcx = abcx=abc , then aaa , bbb and ccc are the factors of xxx .
Example: 6x2y+15x6x^2y + 15x6x2y+15x has two terms; 15x15x15x has the factors 3, 5 and xxx .
[Basic rules of algebra] What are the three basic properties of addition and multiplication used in algebra?
Associative property a+b+c=(a+b)+c=a+(b+c)a + b + c = (a + b) + c = a + (b + c)a+b+c=(a+b)+c=a+(b+c) and abc=(ab)c=a(bc)abc = (ab)c = a(bc)abc=(ab)c=a(bc) Example: (2⋅3)⋅4=2⋅(3⋅4)=24(2 \cdot 3) \cdot 4 = 2 \cdot (3 \cdot 4) = 24(2⋅3)⋅4=2⋅(3⋅4)=24 Commutative property a+b=b+aa + b = b + aa+b=b+a and ab=baab = baab=ba Example: 2⋅7=7⋅2=142 \cdot 7 = 7 \cdot 2 = 142⋅7=7⋅2=14 Distributive property a(b+c)=ab+aca(b + c) = ab + aca(b+c)=ab+ac Example: 3(x+4)=3x+123(x + 4) = 3x + 123(x+4)=3x+12
[Basic rules of algebra] What are expanding brackets and factoring, and how are they related?
Expanding multiplies a bracket out: a(b+c+d)=ab+ac+ada(b + c + d) = ab + ac + ada(b+c+d)=ab+ac+ad . Factoring takes the common part out in front of a bracket: ab+ac=a(b+c)ab + ac = a(b + c)ab+ac=a(b+c) . They are opposite operations; both use the distributive property.
[Basic rules of algebra] Expand (x+a)(x+b)(x + a)(x + b)(x+a)(x+b) .
x2+(a+b)x+ab\boldsymbol{x^2 + (a + b)x + ab}x2+(a+b)x+ab The middle coefficient is the sum of the two constants, the last term their product.Example: (x+3)(x+2)=x2+2x+3x+6=x2+5x+6(x + 3)(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6(x+3)(x+2)=x2+2x+3x+6=x2+5x+6 .
[Basic rules of algebra] Expand (x+a)(bx2+cx+d)(x + a)(bx^2 + cx + d)(x+a)(bx2+cx+d) and collect powers of xxx .
bx3+(c+ab)x2+(d+ac)x+ad\boldsymbol{bx^3 + (c + ab)x^2 + (d + ac)x + ad}bx3+(c+ab)x2+(d+ac)x+ad Distribute: x(bx2+cx+d)+a(bx2+cx+d)=bx3+cx2+dx+abx2+acx+adx(bx^2 + cx + d) + a(bx^2 + cx + d) = bx^3 + cx^2 + dx + abx^2 + acx + adx(bx2+cx+d)+a(bx2+cx+d)=bx3+cx2+dx+abx2+acx+ad ; then group the x2x^2x2 terms and the xxx terms.Example: (x+1)(2x2+3x+4)=2x3+5x2+7x+4(x + 1)(2x^2 + 3x + 4) = 2x^3 + 5x^2 + 7x + 4(x+1)(2x2+3x+4)=2x3+5x2+7x+4 .
[Basic rules of algebra] Solve 7(3+4t)=11(6t−4)7(3 + 4t) = 11(6t - 4)7(3+4t)=11(6t−4) for ttt .
t=6538\boldsymbol{t = \frac{65}{38}}t=3865 Expand: 21+28t=66t−4421 + 28t = 66t - 4421+28t=66t−44 . Move all ttt terms to one side and constants to the other: 21+44=(66−28)t21 + 44 = (66 - 28)t21+44=(66−28)t , so 65=38t65 = 38t65=38t .
[Basic rules of algebra] Factor 6x2y+15x6x^2y + 15x6x2y+15x .
3x(2xy+5)\boldsymbol{3x(2xy + 5)}3x(2xy+5) 6x2y=3⋅2⋅x⋅x⋅y6x^2y = 3 \cdot 2 \cdot x \cdot x \cdot y6x2y=3⋅2⋅x⋅x⋅y and 15x=5⋅3⋅x15x = 5 \cdot 3 \cdot x15x=5⋅3⋅x ; the common factors are 3 and xxx .Example: 2x2y+2x+4x=2x(xy+1+2)=2x(xy+3)2x^2y + 2x + 4x = 2x(xy + 1 + 2) = 2x(xy + 3)2x2y+2x+4x=2x(xy+1+2)=2x(xy+3) .
[Basic rules of algebra] What are the roots of a function fff ?
The inputs xxx with f(x)=0f(x) = 0f(x)=0 . Example: f(x)=x2−5x+6f(x) = x^2 - 5x + 6f(x)=x2−5x+6 has roots x1=2x_1 = 2x1=2 and x2=3x_2 = 3x2=3 .
[Basic rules of algebra] f(x)=x2−5x+6f(x) = x^2 - 5x + 6f(x)=x2−5x+6 . Factor fff , and find for which xxx it is positive and for which negative.
f(x)=(x−2)(x−3)\boldsymbol{f(x) = (x - 2)(x - 3)}f(x)=(x−2)(x−3) : positive for x<2x < 2x<2 and for x>3x > 3x>3 , negative for 2<x<32 < x < 32<x<3 . For x>3x > 3x>3 both factors are positive; for x<2x < 2x<2 both are negative and their product is positive; for 2<x<32 < x < 32<x<3 only x−3x - 3x−3 is negative.
[Basic rules of algebra] What is completing the square for a quadratic expression Ax2+Bx+CAx^2 + Bx + CAx2+Bx+C ?
Rewriting it as A(x−h)2+k\boldsymbol{A(x - h)^2 + k}A(x−h)2+k for constants hhh and kkk . Example: x2+5x+6=(x+2.5)2−14x^2 + 5x + 6 = (x + 2.5)^2 - \frac{1}{4}x2+5x+6=(x+2.5)2−41 , so h=−2.5h = -2.5h=−2.5 , k=−14k = -\frac{1}{4}k=−41 .
[Basic rules of algebra] Ax2+Bx+C=A(x−h)2+kAx^2 + Bx + C = A(x - h)^2 + kAx2+Bx+C=A(x−h)2+k . What do hhh and kkk mean for the graph?
The graph is the graph of Ax2Ax^2Ax2 shifted hhh units to the right and kkk units upward. Example: x2+5x+6=(x+2.5)2−14x^2 + 5x + 6 = (x + 2.5)^2 - \frac{1}{4}x2+5x+6=(x+2.5)2−41 is the graph of x2x^2x2 shifted 2.5 to the left and 14\frac{1}{4}41 down.
[Basic rules of algebra] How do you find hhh and kkk in Ax2+Bx+C=A(x−h)2+kAx^2 + Bx + C = A(x - h)^2 + kAx2+Bx+C=A(x−h)2+k ?
Expand the right side to Ax2−2Ahx+Ah2+kAx^2 - 2Ahx + Ah^2 + kAx2−2Ahx+Ah2+k and match coefficients: the xxx terms give h=−B2A\boldsymbol{h = -\frac{B}{2A}}h=−2AB , the constants give k=C−Ah2\boldsymbol{k = C - Ah^2}k=C−Ah2 . Example: x2−6x+13x^2 - 6x + 13x2−6x+13 : h=3h = 3h=3 , k=13−9=4k = 13 - 9 = 4k=13−9=4 , so x2−6x+13=(x−3)2+4x^2 - 6x + 13 = (x - 3)^2 + 4x2−6x+13=(x−3)2+4 .Example: x2+4x+1x^2 + 4x + 1x2+4x+1 : h=−2h = -2h=−2 , k=1−4=−3k = 1 - 4 = -3k=1−4=−3 , so x2+4x+1=(x+2)2−3x^2 + 4x +…
[Basic rules of algebra] Complete the square: x2+5x+6x^2 + 5x + 6x2+5x+6 .
(x+2.5)2−14\boldsymbol{(x + 2.5)^2 - \frac{1}{4}}(x+2.5)2−41 x2+5x+6=A(x−h)2+k=Ax2−2Ahx+Ah2+kx^2 + 5x + 6 = A(x - h)^2 + k = Ax^2 - 2Ahx + Ah^2 + kx2+5x+6=A(x−h)2+k=Ax2−2Ahx+Ah2+k . The x2x^2x2 terms give A=1A = 1A=1 ; the xxx terms give −2h=5-2h = 5−2h=5 , so h=−2.5h = -2.5h=−2.5 .The constants give k=6−(2.5)2=24−254=−14k = 6 - (2.5)^2 = \frac{24 - 25}{4} = -\frac{1}{4}k=6−(2.5)2=424−25=−41 .
[Solving quadratic equations] What is a quadratic equation, and what is its standard form?
An equation containing the unknown squared. Standard form: ax2+bx+c=0\boldsymbol{ax^2 + bx + c = 0}ax2+bx+c=0 . To reach standard form, move all terms to one side, leaving 0 on the other. The name comes from the Latin quadratus, square.Example: x2=45x+23x^2 = 45x + 23x2=45x+23 becomes x2−45x−23=0x^2 - 45x - 23 = 0x2−45x−23=0 .
[Solving quadratic equations] What are the solutions of ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 (the quadratic formula)?
x1,2=−b±b2−4ac2a\boldsymbol{x_{1,2} = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}}x1,2=2a−b±b2−4ac Example: x2−5x+6=0x^2 - 5x + 6 = 0x2−5x+6=0 : x=5±25−242x = \frac{5 \pm \sqrt{25 - 24}}{2}x=25±25−24 , so x1=3x_1 = 3x1=3 , x2=2x_2 = 2x2=2 .
[Solving quadratic equations] Solve x2=45x+23x^2 = 45x + 23x2=45x+23 .
x=45±21172\boldsymbol{x = \frac{45 \pm \sqrt{2117}}{2}}x=245±2117 , i.e. x1≈45.5054\boldsymbol{x_1 \approx 45.5054}x1≈45.5054 , x2≈−0.5054\boldsymbol{x_2 \approx -0.5054}x2≈−0.5054 Standard form: x2−45x−23=0x^2 - 45x - 23 = 0x2−45x−23=0 , so a=1a = 1a=1 , b=−45b = -45b=−45 , c=−23c = -23c=−23 . b2−4ac=2025+92=2117b^2 - 4ac = 2025 + 92 = 2117b2−4ac=2025+92=2117 , and 2117≈46.0109\sqrt{2117} \approx 46.01092117≈46.0109 .
[Solving quadratic equations] How is the quadratic formula for ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 derived?
Move ccc to the right, divide by aaa , then complete the square on the left and take the square root of both sides. x2+bax=−cax^2 + \frac{b}{a}x = -\frac{c}{a}x2+abx=−ac becomes (x+b2a)2−b24a2=−ca\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2} = -\frac{c}{a}(x+2ab)2−4a2b2=−ac .So (x+b2a)2=b2−4ac4a2\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}(x+2ab)2=4a2b2−4ac , and x+b2a=±b2−4ac2ax + \frac{b}{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{2a}x+2ab=±2ab2−4ac .
[Solving quadratic equations] Deriving the quadratic formula: after dividing ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 by aaa , we complete the square x2+bax=(x−h)2+kx^2 + \frac{b}{a}x = (x - h)^2 + kx2+abx=(x−h)2+k .What are hhh and kkk ?
h=−b2a\boldsymbol{h = -\frac{b}{2a}}h=−2ab and k=−b24a2\boldsymbol{k = -\frac{b^2}{4a^2}}k=−4a2b2 (x−h)2+k=x2−2hx+h2+k(x - h)^2 + k = x^2 - 2hx + h^2 + k(x−h)2+k=x2−2hx+h2+k ; the xxx terms give −2h=ba-2h = \frac{b}{a}−2h=ab , and the constants give h2+k=0h^2 + k = 0h2+k=0 .So x2+bax=(x+b2a)2−b24a2x^2 + \frac{b}{a}x = \left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2}x2+abx=(x+2ab)2−4a2b2 .
[Solving quadratic equations] Deriving the quadratic formula for ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 leads to x+b2a=±−ca+b24a2x + \frac{b}{2a} = \pm\sqrt{-\frac{c}{a} + \frac{b^2}{4a^2}}x+2ab=±−ac+4a2b2 .Simplify −ca+b24a2\sqrt{-\frac{c}{a} + \frac{b^2}{4a^2}}−ac+4a2b2 .
b2−4ac2a\boldsymbol{\dfrac{\sqrt{b^2 - 4ac}}{2a}}2ab2−4ac Put both fractions over 4a24a^24a2 : −4ac4a2+b24a2=b2−4ac4a2-\frac{4ac}{4a^2} + \frac{b^2}{4a^2} = \frac{b^2 - 4ac}{4a^2}−4a24ac+4a2b2=4a2b2−4ac , and 4a2=2a\sqrt{4a^2} = 2a4a2=2a (taking a>0a > 0a>0 ; for a<0a < 0a<0 the ±\pm± absorbs the sign).
[Solving quadratic equations] How else can one prove that x1,2=−b±b2−4ac2ax_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x1,2=2a−b±b2−4ac solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 , without deriving the formula?
Substitute x1x_1x1 and x2x_2x2 into ax2+bx+cax^2 + bx + cax2+bx+c and check that the result is 0.
[Solving quadratic equations] What is the golden ratio φ\varphiφ , and which quadratic equation defines it?
φ=1+52≈1.618\boldsymbol{\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618}φ=21+5≈1.618 , the positive solution of x2−x−1=0\boldsymbol{x^2 - x - 1 = 0}x2−x−1=0 . The other solution is 1−52=−1φ≈−0.618\frac{1 - \sqrt{5}}{2} = -\frac{1}{\varphi} \approx -0.61821−5=−φ1≈−0.618 .Check: 1+52⋅1−52=1−54=−1\frac{1 + \sqrt{5}}{2} \cdot \frac{1 - \sqrt{5}}{2} = \frac{1 - 5}{4} = -121+5⋅21−5=41−5=−1 .
[Solving quadratic equations] A ball is thrown upward from a height of 3 m at 12 m/s. Its time of flight solves −4.9t2+12t+3=0-4.9t^2 + 12t + 3 = 0−4.9t2+12t+3=0 , with solutions t1≈−0.229t_1 \approx -0.229t1≈−0.229 and t2≈2.678t_2 \approx 2.678t2≈2.678 . Which is the answer?
t2≈2.678\boldsymbol{t_2 \approx 2.678}t2≈2.678 s t1<0t_1 < 0t1<0 is a time before the throw, so the context of the problem rules it out. t=−12±144+58.8−9.8=−12±14.24−9.8t = \frac{-12 \pm \sqrt{144 + 58.8}}{-9.8} = \frac{-12 \pm 14.24}{-9.8}t=−9.8−12±144+58.8=−9.8−12±14.24 .
[Solving quadratic equations] x1x_1x1 , x2x_2x2 are the solutions of ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 . How does ax2+bx+cax^2 + bx + cax2+bx+c factor?
ax2+bx+c=a(x−x1)(x−x2)\boldsymbol{ax^2 + bx + c = a(x - x_1)(x - x_2)}ax2+bx+c=a(x−x1)(x−x2) x1x_1x1 , x2x_2x2 are the roots of f(x)=ax2+bx+cf(x) = ax^2 + bx + cf(x)=ax2+bx+c : the points where the graph of fff meets the xxx -axis.Example: 2x2−10x+122x^2 - 10x + 122x2−10x+12 has roots 2, 3, so it equals 2(x−2)(x−3)2(x - 2)(x - 3)2(x−2)(x−3) .
[Solving quadratic equations] How does the sign of b2−4acb^2 - 4acb2−4ac determine how many roots f(x)=ax2+bx+cf(x) = ax^2 + bx + cf(x)=ax2+bx+c has?
b2−4ac>0b^2 - 4ac > 0b2−4ac>0 : two roots The graph crosses the xxx -axis twice. Example: x2−5x+6x^2 - 5x + 6x2−5x+6 : 25−24=125 - 24 = 125−24=1 ; roots 2 and 3. b2−4ac=0b^2 - 4ac = 0b2−4ac=0 : one root The graph touches the xxx -axis at one point. Example: x2−2x+1=(x−1)2x^2 - 2x + 1 = (x - 1)^2x2−2x+1=(x−1)2 : 4−4=04 - 4 = 04−4=0 ; root 1. b2−4ac<0b^2 - 4ac < 0b2−4ac<0 : no roots The graph does not touch the xxx -axis, and fff cannot be factored. Example: x2+1x^2 + 1x2+1 : 0−4=−40 - 4 = -40−4=…
[The Cartesian plane] What is the Cartesian plane?
A graphical representation of pairs of numbers (x,y)(x, y)(x,y) , with a horizontal xxx -axis and a vertical yyy -axis, each marked at regular intervals. Named after René Descartes.
[The Cartesian plane] How do you find the point P=(Px,Py)P = (P_x, P_y)P=(Px,Py) in the Cartesian plane?
Start at the origin (0,0)(0, 0)(0,0) , move PxP_xPx along the xxx -axis, then PyP_yPy along the yyy -axis. Example: P=(−3,2)P = (-3, 2)P=(−3,2) : 3 to the left, 2 up.
[The Cartesian plane] How does a vector v⃗=(vx,vy)\vec{v} = (v_x, v_y)v=(vx,vy) differ from a point (Px,Py)(P_x, P_y)(Px,Py) in the plane?
A vector is an arrow that may start anywhere; a point is located from the origin. Every arrow with the same displacement (vx,vy)(v_x, v_y)(vx,vy) is the same vector. Example: v⃗2\vec{v}_2v2 (from (0,0)(0,0)(0,0) to (−1,−2)(-1,-2)(−1,−2) ) and v⃗3\vec{v}_3v3 (from (−2,0)(-2,0)(−2,0) to (−3,−2)(-3,-2)(−3,−2) ) are the same vector (−1,−2)(-1, -2)(−1,−2) : left 1, down 2.
[The Cartesian plane] What is the graph of a function f:R→Rf: \mathbb{R} \to \mathbb{R}f:R→R ?
The set of points (x,y)=(x,f(x))(x, y) = (x, f(x))(x,y)=(x,f(x)) in the Cartesian plane, one for each input xxx . The height yyy of the graph at xxx is the value f(x)f(x)f(x) . Example: the graph of f(x)=x2f(x) = x^2f(x)=x2 passes through (−2,4)(-2, 4)(−2,4) , (0,0)(0, 0)(0,0) , (1,1)(1, 1)(1,1) , (2,4)(2, 4)(2,4) .
[The Cartesian plane] When plotting y=f(x)y = f(x)y=f(x) , the xxx -axis shows the independent variable and the yyy -axis the dependent variable.
xxx varies freely; f(x)f(x)f(x) depends on xxx .
[Functions] A concert ticket costs 25 dollars, and nnn tickets are sold. What is the revenue R(n)R(n)R(n) , and how many tickets bring 7000 dollars?
R(n)=25n\boldsymbol{R(n) = 25n}R(n)=25n ; 280 tickets. Solve 25n=700025n = 700025n=7000 : n=280n = 280n=280 .
[Functions] Concert tickets cost 25 dollars, a 5% processing charge is taken off the revenue, and hosting the concert costs 2000 dollars. What is the profit P(n)P(n)P(n) for nnn tickets sold?
P(n)=23.75n−2000\boldsymbol{P(n) = 23.75n - 2000}P(n)=23.75n−2000 P(n)=R(n)−CP(n) = R(n) - CP(n)=R(n)−C with revenue R(n)=0.95⋅25⋅n=23.75nR(n) = 0.95 \cdot 25 \cdot n = 23.75nR(n)=0.95⋅25⋅n=23.75n and cost C=2000C = 2000C=2000 .Example: n=100n = 100n=100 gives P=2375−2000=375P = 2375 - 2000 = 375P=2375−2000=375 dollars.
[Functions] What four things should you know about a function to "know" it?
Definition What the function does exactly; from it follow the domain, the range and the inverse function. Graph What the plot of xxx versus f(x)f(x)f(x) looks like. Values The outputs for important inputs. Example: cos(60∘)=12\cos(60^\circ) = \frac{1}{2}cos(60∘)=21 Relations How the function relates to other functions. Example: sin2x+cos2x=1\sin^2 x + \cos^2 x = 1sin2x+cos2x=1
[Functions] What is a function f:A→Bf: A \to Bf:A→B ?
A mapping that assigns to each input xxx in the set AAA an output f(x)f(x)f(x) in the set BBB . We say " fff maps xxx to f(x)f(x)f(x) "; the output is also written y=f(x)y = f(x)y=f(x) . In this book mostly f:R→Rf: \mathbb{R} \to \mathbb{R}f:R→R .
[Functions] For a function f:A→Bf: A \to Bf:A→B , what are its domain, image (range) and codomain?
Domain The set of allowed inputs, AAA . Example: for f(x)=x2f(x) = x^2f(x)=x2 , f:R→Rf: \mathbb{R} \to \mathbb{R}f:R→R : R\mathbb{R}R Image (range) The set of all outputs f(x)f(x)f(x) that actually occur. Example: for f(x)=x2f(x) = x^2f(x)=x2 : [0,∞)={y∈R∣y≥0}[0, \infty) = \{y \in \mathbb{R} \mid y \ge 0\}[0,∞)={y∈R∣y≥0} Codomain The type of outputs, BBB ; it contains the image. Example: for f(x)=x2f(x) = x^2f(x)=x2 , f:R→Rf: \mathbb{R} \to \mathbb{R}f:R→R : R\mathbb{R}R
[Functions] When is a function injective (one-to-one)?
When it maps different inputs to different outputs. Example: f(x)=2xf(x) = 2xf(x)=2x is injective; f(x)=x2f(x) = x^2f(x)=x2 is not, since f(2)=f(−2)=4f(2) = f(-2) = 4f(2)=f(−2)=4 .
[Functions] When is a function surjective (onto)?
When its image equals its codomain: every element of the codomain is the output of some input. Example: f(x)=x2f(x) = x^2f(x)=x2 as a function R→R\mathbb{R} \to \mathbb{R}R→R is not surjective: −1-1−1 is the image of no xxx .
[Functions] When is a function f:X→Yf: X \to Yf:X→Y bijective?
When it is both injective and surjective: for each y∈Yy \in Yy∈Y there is exactly one x∈Xx \in Xx∈X with f(x)=yf(x) = yf(x)=y . "At least one xxx " is the surjective part, "at most one xxx " the injective part. fff is then a one-to-one correspondence between XXX and YYY .Example: f(x)=3x+5f(x) = 3x + 5f(x)=3x+5 , f:R→Rf: \mathbb{R} \to \mathbb{R}f:R→R ; the xxx for yyy is y−53\frac{y - 5}{3}3y−5 .
[Functions] What is the composition f∘gf \circ gf∘g of g:A→Bg: A \to Bg:A→B and f:B→Cf: B \to Cf:B→C ?
The function f∘g:A→Cf \circ g: A \to Cf∘g:A→C with (f∘g)(x)=f(g(x))\boldsymbol{(f \circ g)(x) = f(g(x))}(f∘g)(x)=f(g(x)) : first apply ggg , then fff . Example: g(x)=x+1g(x) = x + 1g(x)=x+1 , f(x)=x2f(x) = x^2f(x)=x2 : (f∘g)(2)=f(3)=9(f \circ g)(2) = f(3) = 9(f∘g)(2)=f(3)=9 , but (g∘f)(2)=g(4)=5(g \circ f)(2) = g(4) = 5(g∘f)(2)=g(4)=5 .
[Functions] Which functions f:A→Bf: A \to Bf:A→B have an inverse function f−1:B→Af^{-1}: B \to Af−1:B→A , and what does it satisfy?
Bijective functions; the inverse satisfies f−1(f(x))=x\boldsymbol{f^{-1}(f(x)) = x}f−1(f(x))=x for every x∈Ax \in Ax∈A . f−1f^{-1}f−1 maps each value f(x)f(x)f(x) back to the xxx it came from.
[Functions] Why must f:A→Bf: A \to Bf:A→B be bijective to have an inverse function f−1:B→Af^{-1}: B \to Af−1:B→A ?
Surjective: every y∈By \in By∈B needs some xxx with f(x)=yf(x) = yf(x)=y ; injective: that xxx must be unique, so f−1(y)f^{-1}(y)f−1(y) is defined for every yyy and has one value. Example: f(x)=x2f(x) = x^2f(x)=x2 on R\mathbb{R}R fails both: f−1(−1)f^{-1}(-1)f−1(−1) has no value, and f−1(4)f^{-1}(4)f−1(4) would be both 2 and −2-2−2 .
[Functions] How are most functions other than +,−,×,÷+, -, \times, \div+,−,×,÷ , roots and powers denoted, and what does cos(60∘)\cos(60^\circ)cos(60∘) mean?
By a name; cos(60∘)\cos(60^\circ)cos(60∘) is the value of the function cos\coscos at the input 60∘60^\circ60∘ , which equals 12\frac{1}{2}21 . For other inputs, e.g. cos(33.13∘)\cos(33.13^\circ)cos(33.13∘) , one uses a calculator.
[Functions] What are three ways, besides the definition, to get a feel for a function?
Table of values List input-output pairs (x,f(x))(x, f(x))(x,f(x)) for some inputs, including x=0,1,−1,100x = 0, 1, -1, 100x=0,1,−1,100 . Example: for f(x)=2xf(x) = 2^xf(x)=2x : (0,1),(1,2),(10,1024),(−1,12)(0, 1), (1, 2), (10, 1024), (-1, \frac{1}{2})(0,1),(1,2),(10,1024),(−1,21) Function graph The points (x,f(x))(x, f(x))(x,f(x)) drawn in the Cartesian plane. Facts and properties Known values and relations to other functions. Example: sin(30∘)=12\sin(30^\circ) = \frac{1}{2}sin(30∘)=21 and s…
[Functions] The horizontal axis of a coordinate system, which measures xxx , is sometimes called the abscissa.
The vertical axis measures f(x)f(x)f(x) , written y≡f(x)y \equiv f(x)y≡f(x) for short.
[Functions] What does the notation sin2(x)\sin^2(x)sin2(x) mean?
(sin(x))2\boldsymbol{(\sin(x))^2}(sin(x))2 Example: sin2(30∘)=(12)2=14\sin^2(30^\circ) = \left(\frac{1}{2}\right)^2 = \frac{1}{4}sin2(30∘)=(21)2=41 .
[Functions] Find cos(30∘)\cos(30^\circ)cos(30∘) from sin(30∘)=12\sin(30^\circ) = \frac{1}{2}sin(30∘)=21 and sin2x+cos2x=1\sin^2 x + \cos^2 x = 1sin2x+cos2x=1 .
cos(30∘)=32\boldsymbol{\cos(30^\circ) = \frac{\sqrt{3}}{2}}cos(30∘)=23 cos2(30∘)=1−14=34\cos^2(30^\circ) = 1 - \frac{1}{4} = \frac{3}{4}cos2(30∘)=1−41=43 ; take the positive square root, since cos(30∘)>0\cos(30^\circ) > 0cos(30∘)>0 .
[Functions] f:R→Rf: \mathbb{R} \to \mathbb{R}f:R→R , f(x)=x2+2x+3f(x) = x^2 + 2x + 3f(x)=x2+2x+3 . Compute f(0)f(0)f(0) , f(1)f(1)f(1) and f(2)f(2)f(2) .
f(0)=3\boldsymbol{f(0) = 3}f(0)=3 , f(1)=6\boldsymbol{f(1) = 6}f(1)=6 , f(2)=11\boldsymbol{f(2) = 11}f(2)=11 f(1)=1+2+3f(1) = 1 + 2 + 3f(1)=1+2+3 , f(2)=4+4+3\ f(2) = 4 + 4 + 3 f(2)=4+4+3 .
[Functions] f(x)=2xf(x) = 2^xf(x)=2x . Which point with x=0x = 0x=0 lies on its graph, and what are f(−1)f(-1)f(−1) , f(−2)f(-2)f(−2) , f(−3)f(-3)f(−3) ?
(0,1)\boldsymbol{(0, 1)}(0,1) ; f(−1)=12\ \boldsymbol{f(-1) = \frac{1}{2}} f(−1)=21 , f(−2)=14\boldsymbol{f(-2) = \frac{1}{4}}f(−2)=41 , f(−3)=18\boldsymbol{f(-3) = \frac{1}{8}}f(−3)=81 Any nonzero number raised to the exponent 0 gives 1; negative exponents give fractions: 2−n=12n2^{-n} = \frac{1}{2^n}2−n=2n1 .Example: f(10)=1024f(10) = 1024f(10)=1024 ; memory chips come in powers of 2 because the memory size is 2(number of address lines)2^{(\text{number of address lines})}2(number of addre…
[Function reference] What is the equation of a line as a function, and what do its two constants mean?
f(x)=mx+bf(x) = mx + bf(x)=mx+b : mmm is the slope, bbb is the yyy -intercept (the value f(0)f(0)f(0) ). Example: f(x)=2x−3f(x) = 2x - 3f(x)=2x−3 has slope 222 and yyy -intercept f(0)=−3f(0) = -3f(0)=−3 .
[Function reference] What input-output relationship does a line f(x)=mx+bf(x) = mx + bf(x)=mx+b describe?
The change in the output is proportional to the change in the input. Changing xxx by Δx\Delta xΔx changes f(x)f(x)f(x) by m Δxm \, \Delta xmΔx .Example: for f(x)=2x−3f(x) = 2x - 3f(x)=2x−3 , increasing xxx by 111 increases f(x)f(x)f(x) by 222 : f(0)=−3f(0) = -3f(0)=−3 , f(1)=−1f(1) = -1f(1)=−1 .
[Function reference] What are the domain and the image of f(x)=mx+bf(x) = mx + bf(x)=mx+b ?
Domain R\mathbb{R}R ; image R\mathbb{R}R if m≠0m \neq 0m=0 , and {b}\{b\}{b} if m=0m = 0m=0 . With m=0m = 0m=0 the function is the constant f(x)=bf(x) = bf(x)=b .Example: f(x)=2x−3f(x) = 2x - 3f(x)=2x−3 takes every real value; f(x)=5f(x) = 5f(x)=5 takes only the value 555 .
[Function reference] What is the xxx -intercept of the line f(x)=mx+bf(x) = mx + bf(x)=mx+b (with m≠0m \neq 0m=0 ), and how is it found?
x=−bm\boldsymbol{x = -\frac{b}{m}}x=−mb , the solution of f(x)=0f(x) = 0f(x)=0 . mx+b=0 ⟺ x=−bmmx + b = 0 \iff x = -\frac{b}{m}mx+b=0⟺x=−mb .Example: 2x−3=02x - 3 = 02x−3=0 gives x=32x = \frac{3}{2}x=23 .
[Function reference] Through two points (x1,y1)(x_1, y_1)(x1,y1) and (x2,y2)(x_2, y_2)(x2,y2) , when does exactly one line f(x)=mx+bf(x) = mx + bf(x)=mx+b pass?
When x1≠x2x_1 \neq x_2x1=x2 . If x1=x2x_1 = x_2x1=x2 and y1≠y2y_1 \neq y_2y1=y2 , no function takes two values at the same input.Example: through (0,−3)(0, -3)(0,−3) and (1,−1)(1, -1)(1,−1) passes only f(x)=2x−3f(x) = 2x - 3f(x)=2x−3 .
[Function reference] What is the inverse of the line f(x)=mx+bf(x) = mx + bf(x)=mx+b (with m≠0m \neq 0m=0 )?
f−1(x)=1m(x−b)\boldsymbol{f^{-1}(x) = \frac{1}{m}(x - b)}f−1(x)=m1(x−b) , which is also a line. Solve y=mx+by = mx + by=mx+b for xxx .Example: f(x)=2x−3f(x) = 2x - 3f(x)=2x−3 has f−1(x)=12(x+3)f^{-1}(x) = \frac{1}{2}(x + 3)f−1(x)=21(x+3) ; f(1)=−1f(1) = -1f(1)=−1 and f−1(−1)=1f^{-1}(-1) = 1f−1(−1)=1 .
[Function reference] What is the general equation of a line?
Ax+By=C\boldsymbol{Ax + By = C}Ax+By=C A relation between xxx and yyy , symmetric in the two variables.Example: the line f(x)=2x−3f(x) = 2x - 3f(x)=2x−3 has general equation 2x−1y=32x - 1y = 32x−1y=3 .
[Function reference] How do you convert the general equation Ax+By=CAx + By = CAx+By=C (with B≠0B \neq 0B=0 ) to the form y=mx+by = mx + by=mx+b ?
m=−AB\boldsymbol{m = -\frac{A}{B}}m=−BA and b=CB\boldsymbol{b = \frac{C}{B}}b=BC Solve for yyy : y=−ABx+CBy = -\frac{A}{B}x + \frac{C}{B}y=−BAx+BC .Example: 2x−1y=32x - 1y = 32x−1y=3 gives m=−2−1=2m = -\frac{2}{-1} = 2m=−−12=2 and b=3−1=−3b = \frac{3}{-1} = -3b=−13=−3 .
[Function reference] The function f(x)=x2f(x) = x^2f(x)=x2 is called the quadratic function, and its graph is a parabola.
"Quadratic" comes from the Latin quadratus, square: a square with side length xxx has area x2x^2x2 .
[Function reference] What are the domain and the image of f(x)=x2f(x) = x^2f(x)=x2 ?
Domain R\mathbb{R}R ; image [0,∞)[0, \infty)[0,∞) . x2≥0x^2 \ge 0x2≥0 for all x∈Rx \in \mathbb{R}x∈R . Example: the graph passes through (−2,4)(-2, 4)(−2,4) , (−1,1)(-1, 1)(−1,1) , (0,0)(0, 0)(0,0) , (1,1)(1, 1)(1,1) , (2,4)(2, 4)(2,4) .
[Function reference] Why is f(x)=x2f(x) = x^2f(x)=x2 called two-to-one?
The inputs xxx and −x-x−x have the same output: x2=(−x)2x^2 = (-x)^2x2=(−x)2 . So every output y>0y > 0y>0 comes from two inputs, y\sqrt{y}y and −y-\sqrt{y}−y .Example: (−3)2=32=9(-3)^2 = 3^2 = 9(−3)2=32=9 .
[Function reference] The function f(x)=x2f(x) = x^2f(x)=x2 is convex: its graph curves upward.
[Function reference] For c≥0c \ge 0c≥0 , what does the symbol c\sqrt{c}c denote?
The nonnegative solution of x2=cx^2 = cx2=c . The other solution is −c-\sqrt{c}−c . Example: x2=9x^2 = 9x2=9 has solutions 333 and −3-3−3 , and 9=3\sqrt{9} = 39=3 .
[Function reference] The square root x=x12\sqrt{x} = x^{\frac{1}{2}}x=x21 is the inverse of which function?
Of x2x^2x2 restricted to x≥0x \ge 0x≥0 . On all of R\mathbb{R}R , x2x^2x2 has no inverse, since xxx and −x-x−x have the same square.Example: 42=4\sqrt{4^2} = 442=4 , but (−4)2=4≠−4\sqrt{(-4)^2} = 4 \neq -4(−4)2=4=−4 .
[Function reference] What are the domain and the image of f(x)=xf(x) = \sqrt{x}f(x)=x , and why is the domain restricted?
Domain [0,∞)[0, \infty)[0,∞) , image [0,∞)[0, \infty)[0,∞) . No real number yyy has a negative square y2y^2y2 , so x\sqrt{x}x is not defined for x<0x < 0x<0 . Example: the graph passes through (1,1)(1, 1)(1,1) , (4,2)(4, 2)(4,2) , (9,3)(9, 3)(9,3) , (16,4)(16, 4)(16,4) .
[Function reference] What are the cube root x3\sqrt[3]{x}3x and the nnn th root xn\sqrt[n]{x}nx ?
x3=x13\sqrt[3]{x} = x^{\frac{1}{3}}3x=x31 is the inverse of x3x^3x3 ; xn\sqrt[n]{x}nx is the inverse of xnx^nxn . For even nnn , xnx^nxn is restricted to x≥0x \ge 0x≥0 , as for the square root.Example: 83=2\sqrt[3]{8} = 238=2 , since 2×2×2=82 \times 2 \times 2 = 82×2×2=8 .