2.0E: Exercises

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Dec 21, 2020
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- 2.0: Library of functions
- 2.1: Introduction to concept of a limit

This page is a draft and is under active development.

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For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

Exercise $\PageIndex{1}$

$x$	$y$	$x$	$y$
-3	9	1	1
-2	4	2	4
-1	1	3	9
0	0

Answer

a. Domain = { $−3,−2,−1,0,1,2,3$ }, range = { $0,1,4,9$ }

b. Yes, a function

Exercise $\PageIndex{2}$

$x$	$y$	$x$	$y$
1	-3	1	1
2	-2	2	2
3	-1	3	3
0	0

Answer

a. Domain = { $0,1,2,3$ }, range = { $−3,−2,−1,0,1,2,3$ }

b. No, not a function

Exercise $\PageIndex{3}$

$x$	$y$	$x$	$y$
3	3	15	1
5	2	21	2
8	1	33	3
10	0

Answer

a. Domain = { $3,5,8,10,15,21,33$ }, range = { $0,1,2,3$ }

b. Yes, a function

For the following exercises, find the values for each function, if they exist, then simplify.

a. $f(0)$ b. $f(1)$ c. $f(3)$ d. $f(−x)$ e. $f(a)$ f. $f(a+h)$

Exercise $\PageIndex{4}$

$f(x)=5x−2$

Answer: a. $−2$ b. $3$ c. $13$ d. $−5x−2$ e. $5a−2$ f. $5a+5h−2$

Exercise $\PageIndex{5}$

$f(x)=\frac{2}{x}$

Answer: a. Undefined b. $2$ c. $23$ d. $−\frac{2}{x}$ e $\frac{2}{a}$ f. $\frac{2}{a+h}$

Exercise $\PageIndex{6}$

$f(x)=\sqrt{6x+5}$

Answer: a. $\sqrt{5}$ b. $\sqrt{11}$ c. $\sqrt{23}$ d. $\sqrt{−6x+5}$ e. $\sqrt{6a+5}$ f. $\sqrt{6a+6h+5}$

Exercise $\PageIndex{7}$

$f(x)=|x−7|+8$

Answer: a. $15$ b. $14$ c. $12$ d. $|x+7|+8$ e. $|a−7|+8$ f. $|a+h−7|+8$

Exercise $\PageIndex{8}$

$f(x)=\frac{x−2}{3x+7}$

Answer: a. \frac{-2}{7} b. $-.1$ c. $\frac{1}{17}$ d. $-\frac{x+2}{-3x+7}$ e $\frac{a−2}{3a+7}$ f. $\frac{a+h−2}{3a+3h+7}$

Exercise $\PageIndex{9}$

$f(x)=9$

Answer: a. 9 b. 9 c. 9 d. 9 e. 9 f. 9

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.

Exercise $\PageIndex{10}$

$g(x)=\sqrt{8x−1}$

Answer: $x≥\frac{1}{8};y≥0;x=\frac{1}{8}$ ; no y-intercept

Exercise $\PageIndex{11}$

$f(x)=−1+\sqrt{x+2}$

Answer: $x≥−2;y≥−1;x=−1;y=−1+\sqrt{2}$

Exercise $\PageIndex{12}$

$g(x)=\frac{3}{x−4}$

Answer: $x≠4;y≠0$ ; no x-intercept; $y=−\frac{3}{4}$

Exercise $\PageIndex{13}$

$g(x)=\sqrt{\frac{7}{x−5}}$

Answer: $x>5;y>0$ ; no intercepts

Exercise $\PageIndex{14}$

$f(x)=\frac{x}{x^2−16}$

Answer: $x≠\pm 4$ ; $x=0,y=0$

Contributors and Attributions

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

Pamini Thangarajah (Mount Royal University, Calgary, Alberta, Canada)

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For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

For the following exercises, find the values for each function, if they exist, then simplify.

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.

Contributors and Attributions

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