Algebra

Absolute values and square roots

12 flashcards · Mathematics · by Athena Papkouf

1. |x| relation to zero

   ∣x∣≥0 and ∣x∣=0⇔x=0|x| \geq 0 \text { and }|x|=0 \Leftrightarrow x=0∣x∣≥0 and ∣x∣=0⇔x=0

2. |x|2

   x2

3. | − x |

   |x| −|x | ≤ x ≤ |x |

4. |x + y|

   ≤ |x| + |y|

5. |x · y |

   |x |·|y|

6. relation to sgn(x)

   x = sgn(x )|x| and |x| = sgn(x)x

7. |f (x )| = g(x)

   g(x)≥0 et {f(x)=g(x) or f(x)=−g(x)g(x) \geq 0 \text { et }\left\{\begin{array}{c} f(x)=g(x) \\ \text { or } \\ f(x)=-g(x) \end{array}\right.g(x)≥0 et ⎩⎨⎧​f(x)=g(x) or f(x)=−g(x)​

8. |f (x )| ≥ g(x)

   {f(x)≥g(x)orf(x)≤−g(x)\left\{\begin{array}{c} f(x) \geq g(x) \\ or \\ f(x) \leq-g(x) \end{array}\right.⎩⎨⎧​f(x)≥g(x)orf(x)≤−g(x)​

9. |f (x )| ≤ g(x)

   {f(x)≤g(x)andf(x)≥−g(x)\left\{\begin{array}{c} f(x) \leq g(x) \\ and \\ f(x) \geq-g(x) \end{array}\right.⎩⎨⎧​f(x)≤g(x)andf(x)≥−g(x)​

10. f(x)=g(x)\sqrt{f(x)}=g(x)f(x) ​=g(x)

    ⇔g(x)≥0 et f(x)=(g(x))2 \Leftrightarrow g(x) \geq 0 \text { et } f(x)=(g(x))^{2}⇔g(x)≥0 et f(x)=(g(x))2

11. f(x)≤g(x)\sqrt{f(x)} \leq g(x) f(x) ​≤g(x)

    g(x)≥0 et f(x)≤(g(x))2g(x) \geq 0 \text { et } f(x) \leq(g(x))^{2}g(x)≥0 et f(x)≤(g(x))2

12. f(x)≥g(x)\sqrt{f(x)} \geq g(x)f(x) ​≥g(x)

    {g(x)&lt;0oug(x)≥0 et f(x)≥(g(x))2\left\{\begin{array}{c} g(x)&lt;0 \\ o u \\ g(x) \geq 0 \text { et } f(x) \geq(g(x))^{2} \end{array}\right.⎩⎨⎧​g(x)<0oug(x)≥0 et f(x)≥(g(x))2​