## Section5.12Additional Practice Related to Functions

### ExercisesExercises

As directed, determine either the specified function value or determine the solution set to the stated equation.

###### 1.

Determine $k(9)$ where $k(x)=3+4x-x^2\text{.}$

Solution

$k(9)=-42$

###### 2.

Determine $y(-3)$ where $y(t)=5-x-x^3\text{.}$

Solution

$y(-3)=35$

###### 3.

Determine the solution set to $m(x)=5$ where $m(x)=3x-7\text{.}$

Solution

The solution set is $\{4\}\text{.}$

###### 4.

Determine the solution set to $p(t)=-3$ where $p(t)=t^2-7t-47\text{.}$

Solution

The solution set is $\{-4,11\}\text{.}$

###### 5.

Determine $f(11)$ where $f(x)=\frac{x-4}{x+3}\text{.}$

Solution

$f(11)=\frac{1}{2}$

###### 6.

Determine the solution set to $w(t)=6$ where $w(t)=\sqrt{3t-9}\text{.}$

Solution

The solution set is $\{15\}\text{.}$

###### 7.

Determine $w(6)$ where $w(t)=\sqrt{3t-9}\text{.}$

Solution

$w(6)=3$

###### 8.

Determine the solution set to $q(x)=12$ where $q(x)=3+\abs{x-1}\text{.}$

Solution

The solution set is $\{-8,10\}\text{.}$

Work with a function presented in graphical form.

###### 9.

Determine each of the following function values based upon the function $k$ shown in FigureĀ 5.12.1.

1. $k(2)$
2. $k(0)$
3. $k(4)$
4. $k(-3)$

Determine the solution set to each of the following equations based upon the function $k$ shown in FigureĀ 5.12.1.

1. $k(x)=1$
2. $k(x)=4$
3. $k(x)=5$
4. $k(x)=6$
Solution
1. $k(2)$ is undefined.
2. $k(0)=-3$
3. $k(4)=4$
4. $k(-3)=-5$
1. The solution set is $\{-1,5\}\text{.}$
2. The solution set is $\{4\}\text{.}$
3. The solution set is $\{3\}\text{.}$
4. The solution set is $\emptyset\text{.}$

Determine the domain of each of the following functions. State each domain using interval notation.

###### 10.

$r(x)=4-\sqrt{8-16x}$

Solution

The domain of $r$ is $\left(-\infty,\frac{1}{2}\right]\text{.}$

###### 11.

$w(t)=\sqrt[5]{6t+42}$

Solution

The domain of $w$ is $(-\infty,\infty)\text{.}$

###### 12.

$f(x)=2x^2-7x+8$

Solution

The domain of $f$ is $(-\infty,\infty)\text{.}$

###### 13.

$q(x)=\frac{y-9}{y+10}$

Solution

The domain of $q$ is $(-\infty,-10) \cup (-10,\infty)\text{.}$

###### 14.

$q(t)=\frac{\sqrt{t-12}}{t-3}$

Solution

The domain of $q$ is $[12,\infty)\text{.}$

###### 15.

$y(x)=\frac{x^2-5x+6}{x^2-8x+12}$

Solution

The domain of $y$ is $(-\infty,2) \cup (2,6) \cup (6,\infty)\text{.}$

Determine the domain and range of functions presented in graphical form.

###### 16.

Determine the domain and range of the function $g$ shown in FigureĀ 5.12.2. State the domain and range using interval notation.

Solution

The domain of $g$ is $(-\infty,1)$and the range is $(-\infty,5]\text{.}$

###### 17.

Determine the domain and range of the function $z$ shown in FigureĀ 5.12.3. State the domain and range using interval notation.

Solution

The domain of $z$ is $(-5,0] \cup (3,6)$and the range is $(-1,4]\text{.}$

Evaluate and simplify each indicated expression.

###### 18.

Determine $f(x+3)$ for the function $f(x)=x^2-x$

Solution

$f(x+3)=x^2+5x+6$

###### 19.

Determine $g(-x)+5$ for the function $g(x)=2x^3-3x^2-6x+2\text{.}$

Solution

$g(-x)+5=-2x^3-3x^2+6x+7$

###### 20.

Determine $3g(4t)$ for the function $\sqrt{t-7}\text{.}$

Solution

$3g(4t)=3\sqrt{4t-7}$

###### 21.

Determine $-2y(t)-7$ for the function $y(t)=-5t^2-11t+3\text{.}$

Solution

$-2y(t)-7=10t^2+22t-13$

###### 22.

Determine $3r(4t-3)$ for the function $r(t)=\frac{3t-5}{t+4}\text{.}$

Solution

$3r(4t-3)=\frac{36t-42}{4t+1}$

###### 23.

Determine $\frac{w(y+5)-w(y-5)}{2}$ for the function $w(y)=4y-7\text{.}$

Solution

$\frac{w(y+5)-w(y-5)}{2}=20$

###### 24.

Evaluate and simplify $(f \circ f)(x)$ where $f(x)=7-x\text{.}$

Solution

$(f \circ f)(x)=x$

###### 25.

Evaluate and simplify $(g \circ f)(x)$ where $f(x)=3x+9$ and $g(x)=5-x^2\text{.}$

Solution

$(g \circ f)(x)=-9x^2-54x-76$

###### 26.

Evaluate and simplify $(f \circ g)(4)$ where $f(x)=3x+9$ and $g(x)=5-x^2\text{.}$

Solution

$(f \circ g)(4)=-24$

###### 27.

Evaluate and simplify $(g \circ g)(8)$ where $g(t)=\frac{3}{t+1}\text{.}$

Solution

$(g \circ g)(8)=\frac{9}{4}$

###### 28.

Evaluate and simplify $(k \circ h)(t)$ where $h(t)=\frac{9t+2}{6}$ and $k(t)=\frac{6t-2}{9}\text{.}$

Solution

$k \circ h)(t)=t$

Determine the difference quotient for each of the following functions. Make sure that you completely simplify each expression.

###### 29.

$f(x)=3x+7$

Solution

The difference quotient is $\frac{f(x+h)-f(x)}{h}=3,\,\,h \neq 0\text{.}$

###### 30.

$g(x)=x^2-5x+1$

Solution

The difference quotient is $\frac{g(x+h)-g(x)}{h}=2x+h-5,\,\, h \neq 0\text{.}$

###### 31.

$k(t)=3-2t^2$

Solution

The difference quotient is $\frac{k(t+h)-k(t)}{h}=-4t-2h,\,\, h \neq 0\text{.}$

###### 32.

$s(t)=-\frac{3}{2t}$

Solution

The difference quotient is $\frac{s(t+h)-s(t)}{h}=\frac{3}{2t^2+2th},\,\, h \neq 0\text{.}$

Determine solution sets to inequalities based upon the graph of a function.

###### 33.

Determine the solution set to $k(x) \geq 1$ based upon the function $k$ shown in FigureĀ 5.12.4. State the solution set using both set-builder notation and interval notation.

Solution

The solution set is $\{x \mid -1 \leq x \lt 0 \text{ or } 2 \lt x \leq 5\}\text{.}$

The solution set is $[-1,0) \cup (2,5]\text{.}$

###### 34.

Determine the solution set to $k(x) \leq 4$ based upon the function $k$ shown in FigureĀ 5.12.5. State the solution set using both set-builder notation and interval notation.

Solution

The solution set is $\{x \mid x \leq 0 \text{ or } x \geq 4\}\text{.}$

The solution set is $(-\infty,0] \cup [4,\infty)\text{.}$

###### 35.

Determine the solution set to $k(x) \gt 5$ based upon the function $k$ shown in FigureĀ 5.12.6. State the solution set using both set-builder notation and interval notation.

Solution

The solution set is $\{\}\text{.}$

The solution set is $\emptyset\text{.}$

###### 36.

Determine the solution set to $k(x) \lt 6$ based upon the function $k$ shown in FigureĀ 5.12.7. State the solution set using both set-builder notation and interval notation.

Solution

The solution set is $\{x \mid x \leq 0 \text{ or } x \gt 2\}\text{.}$

The solution set is $(-\infty,0] \cup (2,\infty)\text{.}$

Draw the inverse of each graphed function.

Solution
###### 38.
Solution

Given the formula for $f(x)\text{,}$ determine the formula for $f^{-1}(x)\text{.}$

###### 39.

$f(x)=4x-19$

Solution

$f^{-1}(x)=\frac{1}{4}x+\frac{19}{4}$

###### 40.

$f(x)=\frac{\sqrt[3]{2x+8}}{2}-6$

Solution

$f^{-1}(x)=\frac{1}{2}(2x+12)^3-4$

###### 41.

$f(x)=\frac{1-2x}{3x+11}$

Solution

$f^{-1}(x)=\frac{11-x}{3x+2}$

The graph of a function named $f$ is given in FigureĀ 5.12.12. Three functions are stated in terms of $f\text{.}$ Graph each of these functions.

###### 42.

$g(x)=\frac{1}{3}f(3x-6)+2$

Solution
###### 43.

$h(x)=\frac{2}{3}f\left(\frac{3}{2}(x+1)\right)-1$

Solution
###### 44.

$k(x)=-f(-(x+2))$

Solution

One point on a function named $f$ is $(12,-20)\text{.}$ Several functions are defined in terms of $f\text{.}$ For each function, use graphical transformation properties to determine where the stated point on $f$ resides on the graph of the new function.

###### 45.

$g(x)=-f(4(x-3))-8$

Solution

The point $(12,-20)$ for $f$ moves to $(6,12)$ on $g$

###### 46.

$h(x)=\frac{1}{10}f(\frac{1}{5}x+3)$

Solution

The point $(12,-20)$ for $f$ moves to $(45,-2)$ on $h$

###### 47.

$k(x)=2f(-6x-12)+5$

Solution

The point $(12,-20)$ for $f$ moves to $(-4,-35)$ on $k$

###### 48.

$w(x)=-\frac{2}{5}f(\frac{2}{3}(x+8))-3$

Solution

The point $(12,-20)$ for $f$ moves to $(10,5)$ on $w$