Essential Mathematics: Functions

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95 flashcards · Mathematics · by lakesare

How to prove whether some number is irrational, e.g. −√2, √2, e and π?
It's not uniform - easy to do for √ 2, more difficult for e and π. To check that the numbers − √ 2, √ 2, e and π are irrational, you have to prove that they can’t be written as an integer divided by an integer.If you'd like to see how this can be done for √ 2, then look at the document A proof that √ 2 is irrational on the module website. Proving that e and π can’t be written as an integer divided by an integer is more difficult, and outside the scope of this module.
What are 2 decimal forms of rational numbers?
Terminating It has only a finite number of digits after the decimal point.Recurring Each of them has a block of one or more digits after the decimal point that repeats indefinitely. If the number is rational - its decimal form is always either terminating or recurring. Also, every terminating or recurring decimal can be written as an integer divided by an integer and is therefore a rational number. The irrational numbers are the decimal numbers with an infinite number of digits after the decima…
What is the decimal expansion of irrational numbers?
They are infinitely long, but have no recurrence, - there is no block of digits that repeats indefinitely. The decimal numbers that are neither terminating nor recurring – that is, those that are infinitely long but have no block of digits that repeats indefinitely – are the irrational numbers.
Round the following numbers. (a) 0.0238 to three decimal places (b) 50629 to three significant figures (c) 0.0029582 to two significant figures
(a) 0.0238 = 0.024 (to 3 d.p.) (b) 50629 = 50600 (to 3 s.f.) (c) 0.0029582 = 0.0030 (to 2 s.f.) Notice that in Example 2(c),a0is included after the 3 to make it clear that the number is rounded to two significant figures. You should do likewise when you round numbers yourself.
Round −0.25 to one decimal place.
−0.25 = −0.3 (to 1 d.p.).
Name 3 rules for identifying significant figures.
1. Nonzero numbers and zeros between them are always significant. E.g., 101.1203 has seven significant figures: 1, 0, 1, 1, 2, 0 and 3.2. Zeros to the left of the non-zero digits are not significant. E.g., 0.00052 has two significant figures: 5 and 2.3. Zeros to the right of the non-zero digits are: If after the decimal point - significant, as these are only necessary to indicate precision. E.g., 1.20 and 0.0980 have three significant figures.If before the decimal point - depends. Trailing zero…
The correct answer is 470cm2. We should only round in the last step, - prior to that we should use the most precise values. If you obtained the answer 480 cm2 , then this was probably because you carried out your calculation in part (b) using the rounded answer for the radius that you found in part (a).To obtain the correct answer in part (b), you need to use a more precise value for the radius, such as the full-calculator-precision version.
For the integer 28 - find all its positive factor pairs, and list all its positive factors in increasing order. A factor pair of an integer is a pair of its factors that multiply together to give the integer. For example, the factor pairs of 12 are (1, 12), (2, 6), (3, 4), (−1, −12), (−2, −6), (−3, −4).
The positive factor pairs of 28 are (1, 28), (2, 14), (4, 7). The positive factors of 28 are 1, 2, 4, 7, 14, 28. To find the positive factor pairs of a positive integer• Try dividing the integer by each of the numbers 1, 2, 3, 4,... in turn. Whenever you find a factor, write it down along with the other factor in the factor pair.• Stop when you get a factor pair that you have already.
Describe the algorithm to find the prime factorization of a number greater than 1.
Repeatedly 'factor out' the prime 2 until you obtain a number that isn't divisible by 2. Repeatedly factor out the prime 3 until you obtain a number that isn't divisible by 3. Repeatedly factor out the prime 5 until you obtain a number that isn't divisible by 5. Continue this process with each of the successive primes, 2, 3, 5, 7, 11, 13, 17, ..., in turn. Stop when you have a product of primes. Write out the prime factorisation with the prime factors in increasing order, using index notation f…
What do LCM and HCF stand for?
LCM - Lowest Common Multiple HCF - Highest Common Factor Also known as GCD - Greatest Common Divisor
What is the LCM (Lowest Common Multiple) of a8b and ac?
a8bc, because we should take the highest out of 2 powers (including 0 powers).
What is the HCF (Highest Common Factor) of a8b and ac? Also known as GCD - Greatest Common Divisor
a, because we should take the lowest out of 2 powers (including 0 powers).
What's the strategy to find the LCM (Lowest Common Multiple), and HCF (Highest Common Factor) of two or more integers?
Find the prime factorisations of the numbers. To find the LCM, multiply together the highest power of each prime factor occurring in any of the numbers. To find the HCF, multiply together the lowest power of each prime factor common to all the numbers. There's also an efficient method, known as Euclid's algorithm, for finding lowest common multiples and highest common factors, without having to find prime factorisations first.
Find the LCM (Lowest Common Multiple) and HCF (Highest Common Factor) of 9, 18 and 24 using their prime factorisation.
LCM of 9, 18 and 24: 23 × 32 = 72. HCF of 9, 18 and 24: 31 = 3.
What are 3 different things letters can represent in algebraic expressions? An expression that contains letters (usually as well as numbers and mathematical symbols) is an algebraic expression. An expression, such as 10 − 2 × 4, that contains only numbers and mathematical symbols is a numerical expression.
variable - representing any number of a particular kind. unknown - a particular number that you don't know, and usually want to discover. constant - a particular number whose value is specified. Letters representing numbers in algebraic expressions can be any of three different types. A letter may be a variable: this means that it represents any number, or any number of a particular kind, such as any positive number, or any integer. Alternatively, it may be an unknown: this means that it repres…
Solve and explain why.
Solve and explain why.
Solve and explain why.
What's 00?
Usually we don't give a meaning to 00, but in some contexts it's convenient to take it to be equal to 1. There's no obvious, correct meaning for 00. This is because on the one hand you’d expect that if you raise a number to the power 0 then the answer will be 1, but on the other hand you’d expect that if you raise 0 to a power then the answer will be 0. Mathematicians have debated the meaning of 00 for several centuries, and in 1821 Augustin-Louis Cauchy included it in a list of undefined forms…
Express 427 in scientific notation.
4.27 * 102 In computer output, scientific notation is sometimes represented with the power of 10 indicated by the letter E (for exponent).For example, 4.27 × 10−2 would be represented as 4.27E−2.
What's the strategy for finding a gradient of a line?
You select any 2 points on a line (doesn't matter which points and in what order). Find run: the number of units by which x increases as you go from 1st point to 2nd point. Find rise: the number of units by which y increases as you go from 1st point to 2nd point. If x or y (or both) actually decreases as you trace your pen tip from the first point to the second point, then the run or rise (or both) is negative.
What's the gradient of the horizontal and vertical lines?
horizontal line - gradient is 0. The gradient of a horizontal line is zero. This is because the gradient is the rise divided by the run, and the rise between any two points on a horizontal line is zero. vertical line - gradient is undefined. On the other hand, the gradient of a vertical line is undefined. This is because, again, the gradient is the rise divided by the run, but the run between any two points on a vertical line is zero. Since it is not possible to divide by zero, the gradient of…
What is the only line that doesn't have the equation in the form of y = mx + c?
x = some constant
Why if the gradient of one line is p/q - the gradient of the perpendicular line is -q/p?
It can be seen from analyzing the graph where we rotate the right-angled triangle with hypotenuse AB anticlockwise through a 1/4 turn about A, then it will lie exactly on top of the right-angled triangle with hypotenuse AC.
What's factorisation of a quadratic expression VS completing the square?
Factorisation - is turning a quadratic 2x2 − x − 6 into (2x + 3)(x − 2) or (x − 2)(2x + 3). Completing the square - is turning a quadratic 2x2 − 12x + 25 into 2(x − 3)2 + 7. Many quadratics with integer coefficients can’t be factorised using integers. If you take any two linear expressions and multiply them together, then you get a quadratic expression. For example, (2x + 3)(x − 2) = 2x2 − x − 6. Factorisation is the opposite of this process.
What 2 things are needed to define a function?
To specify a function, a domain must be stated, and a rule. Two functions with the same rule but different domains are different functions.
What's the domain of f(x)?
The domain of f is the set of all real numbers except 4, that is, the set:
Which of the following are the graphs of functions?
(a) - function (b) - no (e) - no (f) - function
Which graphs show functions that are increasing on their whole domains?
(a) - increasing (b) - increasing (c) For the function in graph (c), if you take x1 and x2 to be values slightly less than 0 and slightly greater than 0, respectively, then the function takes a smaller value at x2 than it does at x1, so it is not increasing on its whole domain.(d) For the function in graph (d), you can find values x1 and x2 with x1 < x2 such that the function takes the same value at x2 as it does at x1, so it is not increasing on its whole domain.
How can you visualize the domain and image of the function?
What's the degree of the polynomial?
The highest power of the variable x in a polynomial expression. Quadratic functions are polynomial functions of degree 2, since the highest power of x in the rule of a quadratic function is x2.
What's the degree of this polynomial function: f(x) = ax + b? Where a ≠ 0
Highest power of x is x1, so the degree is 1.
What's the degree of this polynomial function: f(x) = c? Where c ≠ 0
Highest power of x is x0, so the degree is 0. The particular linear function f(x)=0 is usually regarded as not having a degree at all (or sometimes as having degree −∞), for technical reasons.
How to determine whether polynomial function tends to -∞ or ∞ on the left and the right? f(x) = 2x4 − 5x3 + x2 + 2x − 2 The polynomial graph often has a 'wiggly' section, but (unless the function is a constant function) if you trace your pen tip along the graph towards the right, then eventually the wiggles stop and your pen either keeps moving ‘uphill’ or keeps moving 'downhill'. The same happens if you trace your pen tip towards the left.
Look at the dominant term. If the dominant term: contains an even power of x, then the graph tends to ∞ at both ends contains an odd power of x, then the graph tends to -∞ at the left and to ∞ at the right. If the dominant term has a minus sign, then similar facts hold, but with infinity replaced by minus infinity and vice versa. You can tell whether the graph of a polynomial function tends to infinity or tends to minus infinity at each end by looking at the term in its rule that has the highes…
What function is this a graph of?
Modulus function: f(x) = |x|
What function is this a graph of?
Reciprocal function:
What's a rational function?
Function that can be represented as a fraction of 2 polynomials. p(x) and q(x) are polynomials.
[graph translation] Match up the equations with their graphs.
To where is the original graph translated?
The original graph is translated to the right by c units up by d units.
What function is this a graph of?
Exponential function: f(x) = bx
What happens every time you: increase the value of x by 1? decrease the value of x by 1?
Each time x is increased by 1, the value of f(x) doubles. Notice also that each time the value of x increases by 1 unit to the next integer up, the value of f(x) doubles. So as x takes values that are further and further along the number line to the right, the value of f(x)=2x increases, and increases more and more rapidly. This explains the shape of the graph as x increases. Each time x is decreased by 1, the value of f(x) halves (but never reaches 0). Similarly, each time the value of x decre…
What value of b in y = bx gives the flattest graph?
y = 1x
How can you explain the fact that the graph of y = 0.5x reflects the graph of y = 2x across the y-axis?
What should be the value of b in y = bx for gradient at (0, 1) to be exactly 1?
b = 2.71828... (also denoted as e) The graph should go up by the same distance vertically as it goes along horizontally to the right.The use of the letter e for the base of the exponential function was introduced by Leonhard Euler.
What special property does the exponential function with the rule f(x) = ex have? This function is so important that it's sometimes referred to as The exponential function.
f(x) = ex gradient at (0, 1) is exactly 1.
Solve without using the calculator:
Why is ln (loge) named so?
It stands for logarithm natural. Logarithms to base e were first described as ‘natural’ logarithms by the Danish mathematician Nicolaus Mercator (1620–87), in his treatise Logarithmotechnica published in 1668. Mercator’s treatise was difficult to follow, so later writers tended to refer to the exposition of it given by the English mathematician John Wallis (1616–1703), published in the same year. The first published use of the notation ‘ln’ for a natural logarithm was in a book written by an Am…
This is a graph of y = ex. How does the graph of y = ln(x) look?
f(x) = ln(x) is the inverse function of the function g(x) = ex , Which means they are reflections of each other in the line y = x. (provided that the coordinate axes have equal scales)
How could you obtain the graph of h(x) by applying scaling, and applying translations to the graph of the function f(x) = x3.
[Deal with x] Translate to the left by 3 units [Deal with y] Scale y by 1/2 Translate down by 2 units
How could you obtain the graph of j(x) by applying scaling, and applying translations to the graph of the function f(x) = |x|.
[Deal with x] Translate to the right by 3 units [Deal with y] Scale y by 1/2 Translate down by 4 units [Note that only order of dealing with y is important] Scale it vertically by the factor 1/2 , translate it to the right by 3 units, and finally translate it down by 4 units.You can carry out the operations in any order, except that you have to do the vertical scaling before the vertical translation.
Consider the quadratic function f(x) = 2x2 + 12x + 19. Complete the square in the quadratic expression on the right-hand side. Describe how you could obtain the graph of f(x) by applying scalings and translations to the graph of the function f(x) = x2.
1. Completing the square gives: 2. To obtain f(x) from f(x) = x2: [Deal with x] Translate to the left by 3 units [Deal with y] Scale y by 2 units Translate up by 1 unit You can see that this method can be applied to any quadratic function.You just need to start by completing the square in the quadratic expression that forms the right-hand side of its rule. This tells you the following enlightening fact: the graph of any quadratic function is the same basic shape as the graph of y = x2 , but sca…
Given the domains of f(x) and g(x), what's the domain of these combinations of f(x) and g(x)?
The domain of any combination of functions above is: intersection of the domains of f and domain of g with the additional requirement that for the first quotient of f and g, the numbers x such that g(x) = 0 are removed, since it's not possible to divide by zero, and for the second quotient of f and g, the numbers x such that f(x) = 0 are removed. There isn’t much more to be said about sums, products, differences and quotients of functions, at this stage. It’s usually not easy to deduce the shap…
What's the image of 3 under g ⚬ f, where g(x) = x + 1, and f(x) = x2?
g ⚬ f = g(f(x)) = g(32) = 9 + 1 = 10.
Why in g ◦ f, f is applied first (but written second)?
(g ◦ f)(x) = g(f(x)) - we just keep the order. It would be confusing if f and g were in different orders on the two sides of the equation.
Write down the rules of the inverse functions of the following functions:
f -1 (f(x)) = x
Can you think of a function with an inverse function that has the same rule as the original function? Can you think of another such function? f -1 (f(x)) = x
Some possible answers are f(x) = x f(x) = −x f(x) = 1/x There are many others, such as f(x) = 3 − x and f(x) = 8/x.
Give the example of a function that doesn't have an inverse.
f(x) = x2 f(x) is not one-to-one: Therefore the potential f-1(x) is not a function:
What's an easy visual way to recognize that a function is one-to-one?
If a function is one-to-one - horizontal line should cross only once.
Which of the following functions are one-to-one? For each function that isn’t one-to-one, state two input numbers that have the same output number.
If a function is one-to-one - horizontal line should cross only once.
State which of these graphs: is a one-to-one function, not a one-to-one function, or not even a function.
g, l - not a function c, d, j - not one-to-one Others are graphs of one-to-one functions.
What functions have an inverse?
A function has an inverse if and only if it's: it's onto it's one-to-one A function is invertible if and only if it is a bijection.
Find the inverse of the function f(x) = 2x + 1.
To find the rule of f−1, rearrange the equation f(x) = y to express x in terms of y.
Find f-1(x) and its domain.
Why does this trick work: If a function is either increasing or decreasing on its whole domain, then it is one-to-one and so has an inverse function.
If a function increases on its whole domain, then any horizontal line drawn on its graph will cross the graph at most once.
Find the inverse of f(x) = x2 − 4x +1, x ∈ [−1, 1].
1. Try to find the rule of f−1 in the usual way, by rearranging the equation f(x) = y. For a quadratic function like f, it helps to begin by completing the square. 2. Decide whether the + or the − applies. Remember that the final equation above is a rearrangement of the equation f(x) = y, so x is an element of the domain of f, which is [−1, 1]. Now 2 plus the positive square root of something can’t be equal to a number in this interval, but 2 minus the positive square root of something can, so…
How to scale the graph: horizontally vertically? Horizontal scaling Vertical scaling
To scale horizontally: y = f(x/c) vertically: y = cf(x) Informally, when you scale a graph horizontally by a factor of c, you stretch it parallel to the x-axis, and if c is negative, you also reflect it in the y-axis.
[reflections of graphs in the coordinate axes] How to obtain the graph of y = −f(x)?
Reflect the graph of y = f(x) in the x-axis.
[reflections of graphs in the coordinate axes] How to obtain the graph of y = f(−x)?
Reflect the graph of y = f(x) in the y-axis.
What to do when we want to get ~some kind of~ inverse of f(x) = x2? As you know, functions that are not one-to-one don't have inverses.
We 'restrict' the domain of f(x). There are many ways to restrict a function, we choose the one that's comfortable for us. We choose the new domain to make sure that the following two conditions are satisfied: the new function is one-to-one and therefore has an inverse. the image set of the new function is the same as the image set of the original function.
What's the graph of y = ex and y = ln x?
Simplify
Simplify
How to go from bmbn = bm+n to logbx + logby = logb(xy)?
blog(b)x + log(b)y = blog(b)xblog(b)y = xy Which means that logbx + logby = logb(xy) by the definition of log.
Write as a logarithm of a single number: ln 5 + ln 3.
ln 5 + ln 3 = ln(5 × 3) = ln 15
Write as a logarithm of a single number: ln 3 + ln 4 − ln 6.
Simplify: ln c3 − ln c.
ln c3 − ln c = 3 ln c - ln c = 2 ln c
[working with calculator; significant figures] Solve the equation 2 × 1.53x = 45, giving the solution to three significant figures.
Show how any exponential function f(x) = bx can be written as f(x) = ekx. Exponential function is a function in the form f(x) = bx.
By having k = logeb.
What's enlightening about the fact that any exponential function f(x) = bx can be expressed as f(x) = ekx?
It means that the graph of every exponential function is a horizontal scaling of the graph of the exponential function f(x) = ex. It follows that the graphs of all logarithmic functions are vertical scalings of each other, since the graphs of logarithmic functions are reflections of the graphs of exponential functions in the line y = x.
Show the formula for a quantity x that 'changes exponentially'.
f(x) = aekx (or, equivalently, abx). Here is the form of so called exponential models: f(x) = abx.From what you saw in the previous subsection, rules of this form are the same as rules of the form f(x) = aekx, where a and k are non-zero numbers. We’ll use this alternative form in this subsection.
How does the graph of exponential growth look? How does the graph of exponential decay look?
exponential growth (positive k in f(x) = aekx) exponential decay (negative k in f(x) = aekx)
What does this graph show, - exponential growth, or exponential decay?
An exponential decay curve modelling the concentration of a particular prescription drug in a patient’s bloodstream.
When is it okay to × or / both sides of the inequality by a variable?
Only when we know that the variable takes only positive, or only negative values, - because then we know whether we should reverse the inequality sign. When you're rearranging an inequality, you should not multiply or divide both sides by a variable, or by an expression containing a variable, unless you know that the variable or expression takes only positive values or takes only negative values.
Given the inequality and its graph - estimate the solution.
x ∈ [−2, 1] ∪ [3/2, ∞), because these are the ranges where left side is less than or equal to right side. Note that if you solve this in a traditional way, the final graph will offer the same solution:
How is the quadratic formula obtained?
From completing the square of ax2 + bx + c = 0. We can solve any equation by completing the square, - this is just a shortcut.
[table of signs, undefined fraction] Solve this inequality. When you’re trying to solve an inequality that contains an algebraic fraction, remember that you’re not allowed to multiply it through by a variable or expression, unless you know that the variable or expression takes only positive values or takes only negative values. That’s because otherwise you can’t follow the rule about when to reverse the inequality sign.
Suppose f(x) = aekx. What's meant by the growth factor?
Exponential growth functions have an interesting property: f(x)/f(x + p) is always the same for the same p (no matter from what x we start).Growth factor (for a particular p!) is this f(x)/f(x + p) ratio. f(x)/f(x + p) = ekp.Exponential growth and decay functions have an interesting characteristic property. If f is such a function, and you start with any value of x and add a number to it, then the value of f(x) is multiplied by a factor. This factor doesn't depend on the value of x that you sta…
Suppose f(x) = aekx. What's the formula for the growth factor?
f(x)/f(x + p) = ekp.
Suppose that the number of trees of a particular variety in a region, at time t (in decades) after the variety was introduced, is modelled by the exponential growth function: f(t) = 700e0.06t (10 ≤ t ≤ 50). By what factor is the size of the tree population predicted to multiply (a) every decade? (b) every century? (c) every five years?
Name 2 other ways a doubling/halving period may be called.
doubling/halving time half-life In many exponential models, the input variable represents time. When this is the case, the doubling or halving period is more usually called the doubling or halving time. In the particular case of radioactive decay, the halving time is usually called the half-life.
Suppose f(t) = 3e0.4t. What's the doubling period for f(t)?
Such p that e0.4p = 2.