2.E: Graphing Functions and Inequalities (Exercises)

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Exercise $\PageIndex{1}$

Determine the domain and range and state whether the function is a relation or not.

1. $\{ ( - 4 , - 1 ) , ( - 5,3 ) , ( 10,3 ) , ( 11,2 ) , ( 15,1 ) \}$

2. $\{ ( - 3,0 ) , ( - 2,1 ) , ( 1,3 ) , ( 2,7 ) , ( 2,5 ) \}$

Answer

1. Domain: $\{ - 5 , - 4,10,11,15 \}$; range: $\{ - 1,1,2,3 \}$; function: yes

3. Domain: $\{ - 5,5,15,30 \}$; range: $\{ - 5,0,5,10,15 \}$; function: no

5. Domain: $( - \infty , \infty )$; range: $[ - 6 , \infty )$; function: yes

7. Domain: $\left( - \infty , \frac { 3 } { 2 } \right]$; range: $[ 1 , \infty )$; function yes

Exercise $\PageIndex{2}$

Evaluate.

$h (x) = 1 2 x − 3; h (−8), h (3),$ and $h (4a + 1)$
$p (x) = 4 − x ; p (−10), p (0),$ and $p (5a − 1)$
$f (x) = 2x^{2} − x + 3$; find $f (−5), f (0)$, and $f (x + h)$
$g (x) = x^{2} − 9$; find $f (−3), f (2)$, and $f (x + h)$
$g (x) = \sqrt{2x − 1}$; find $g (5), g (1), g (13)$
$h ( x ) = \sqrt [ 3 ] { x + 6 }$; find $h (−7), h (−6)$, and $h (21)$
$f (x) = 8x + 3$; find $x$ where $f (x) = 10$.
$g (x) = 5 − 3x$; find $x$ where $g (x) = −4$.
Given the graph of $f (x)$ below, find $f (−60) , f (0)$, and $f (20)$.

10. . Given the graph of $g (x)$ below, find $x$ where $g (x) = −4$ and $g (x) = 12$.

Answer

1. $h (−8) = −7, h (3) = −\frac{3}{2}$, and $h (4a + 1) = 2a −\frac{5}{2}$

3. $f (−5) = 58, f (0) = 3$, and $f (x + h) = 2x^{2} + 4xh + 2h^{2} − x − h + 3$

5. $g (5) = 3, g (1) = 1, g (13) = 5$

7. $f \left( \frac { 7 } { 8 } \right) = 10$

9. $f ( - 60 ) = - 20 , f ( 0 ) = 20 , f ( 20 ) = 0$

Exercise $\PageIndex{3}$

Graph and label the intercepts.

$4x − 8y = 12$
$9x + 4y = 6$
$\frac{3}{8} x + \frac{1}{2} y = \frac{5}{4}$
$\frac{3}{4} x − \frac{1}{2} y = −1$

Answer

Exercise $\PageIndex{4}$

Graph the linear function and label the $x$-intercept.

$g ( x ) = \frac { 5 } { 8 } x + 10$
$g ( x ) = - \frac { 1 } { 5 } x - 3$
$f ( x ) = - 4 x + \frac { 1 } { 2 }$
$f ( x ) = 3 x - 5$
$h ( x ) = - \frac { 2 } { 3 } x$
$h ( x ) = - 6$

Answer

Exercise $\PageIndex{5}$

Find the slope of the line passing through the given points.

$(−5, 3)$ and $(−4, 1)$
$(7, −8)$ and $(−9, −2)$
$(−\frac{4}{5}, \frac{1}{3})$ and $(−\frac{1}{10}, −\frac{3}{5})$
$(\frac{3}{8}, −1)$ and $(−\frac{3}{4}, −\frac{1}{16})$
$(−14, 7)$ and $(−10, 7)$
$(6, −5)$ and $(6, −2)$

Answer

1. $m=-2$

3. $m=-\frac{4}{3}$

5. $m=0$

Exercise $\PageIndex{6}$

Graph $f$ and $g$ on the same rectangular coordinate plane. Use the graph to find all values of $x$ for which the given relation is true. Verify your answer algebraically.

$f ( x ) = \frac { 1 } { 2 } x - 2 , g ( x ) = - \frac { 5 } { 2 } x + 4 ; f ( x ) = g ( x )$
$f ( x ) = 5 x - 2 , g ( x ) = 3 ; f ( x ) \geq g ( x )$
$f ( x ) = - 4 x + 3 , g ( x ) = - x + 6 ; f ( x ) < g ( x )$
$f ( x ) = \frac { 3 } { 5 } x - 1 , g ( x ) = - \frac { 3 } { 5 } x + 5 ; f ( x ) \leq g ( x )$

Answer

1. $x=2$

3. $( - 1 , \infty )$

Exercise $\PageIndex{7}$

Find the linear function passing through the given points.

$(1, −5)$ and $(\frac{1}{2}, −4)$
$(\frac{5}{3}, −3)$ and $(−2, 8)$
$(7, −6)$ and $(5, −7)$
$(−5, −6)$ and $(−3, −9)$
Find the equation of the given linear function:

6. Find the equation of the given linear function:

Answer

1. $f ( x ) = - 2 x - 3$

3. $f ( x ) = \frac { 1 } { 2 } x - \frac { 19 } { 2 }$

5. $f ( x ) = - \frac { 3 } { 7 } x - \frac { 10 } { 7 }$

Exercise $\PageIndex{8}$

Find the equation of the line:

Parallel to $8x − 3y = 24$ and passing through $(−9, 4)$.
Parallel to $6x + 2y = 24$ and passing through $(\frac{1}{2}, −2)$.
Parallel to $\frac{1}{4} x −\frac{2}{3} y = 1$ and passing through $(−4, −1)$.
Perpendicular to $14x + 7y = 10$ and passing through $(8, −3)$.
Perpendicular to $15x − 3y = 6$ and passing through $(−3, 1)$.
Perpendicular to $\frac { 2 } { 9 } x + \frac { 4 } { 3 } y = \frac { 1 } { 2 }$ and passing through $(2, −7)$.

Answer

1. $y = \frac{8}{3} x + 28$

3. $y = \frac{3}{8} x − \frac{5}{2}$

5. $y = −\frac{1}{5} x + \frac{2}{5}$

Exercise $\PageIndex{9}$

Use algebra to solve the following.

A taxi fare in a certain city includes an initial charge of $$2.50$ plus $$2.00$ per mile driven. Write a function that gives the cost of a taxi ride in terms of the number of miles driven. Use the function to determine the number of miles driven if the total fare is $$9.70$.
A salesperson earns a base salary of $$1,800$ per month and $4.2$% commission on her total sales for that month. Write a function that gives her monthly salary based on her total sales. Use the function to determine the amount of sales for a month in which her salary was $$4,824$.
A certain automobile sold for $$1,200$ in $1980$, after which it began to be considered a collector’s item. In $1994$, the same automobile sold at auction for $$5,750$. Write a linear function that models the value of the automobile in terms of the number of years since $1980$. Use it to estimate the value of the automobile in the year $2000$.
A specialized industrial robot was purchased new for $$62,400$. It has a lifespan of $12$ years, after which it will be considered worthless. Write a linear function that models the value of the robot. Use the function to determine its value after $8$ years of operation.
In $1950$, the U.S. Census Bureau estimated the population of Detroit, MI to be $1.8$ million people. In $1990$, the population was estimated to have decreased to $1$ million. Write a linear function that gives the population of Detroit in millions of people, in terms of years since $1950$. Use the function to estimate the year in which the population decreased to $700,000$ people.
Online sales of a particular product are related to the number of clicks on its advertisement. It was found that $100$ clicks in a week result in $$112$ of online sales, and that $500$ clicks result in $$160$ of online sales. Write a linear function that models the online sales of the product based on the number of clicks on its advertisement. How many clicks are needed to result in $$250$ of weekly online sales from this product?
The cost in dollars of producing n bicycles is given by the formula $C (n) = 80n + 3,380$. If each bicycle can be sold for $$132$, write a function that gives the profit generated by producing and selling $n$ bicycles. Use the formula to determine the number of bicycles that must be produced and sold to profit at least $$10,000$.
Determine the breakeven point from the previous exercise.

Answer

1. $C (x) = 2x + 2.5 ; 3.6$ miles

3. $V (t) = 325t + 1, 200 ; $7,700$

5. $p (x) = −0.02x + 1.8 ; 2005$

7. $P (n) = 52n − 3, 380 ; 258$ bicycles

Exercise $\PageIndex{10}$

Find the ordered pair that specifies the point $P$.

Answer

1. $\left( \frac { 3 } { 2 } , \frac { 2 } { 3 } \right)$

3. $( - 25,25 )$

Exercise $\PageIndex{11}$

Graph the piecewise defined functions.

$g ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } } & { \text { if } x < 5 } \\ { 10 } & { \text { if } x \geq 5 } \end{array} \right.$
$g ( x ) = \left\{ \begin{array} { l l } { - 5 } & { \text { if } x < - 5 } \\ { | x | } & { \text { if } x \geq - 5 } \end{array} \right.$
$f ( x ) = \left\{ \begin{array} { l l } { x } & { \text { if } x \leq - 1 } \\ { x ^ { 3 } } & { \text { if } x > - 1 } \end{array} \right.$
$f ( x ) = \left\{ \begin{array} { l l } { x } & { \text { if } x \leq 4 } \\ { \sqrt { x } } & { \text { if } x > 4 } \end{array} \right.$
$h ( x ) = \left\{ \begin{array} { l l } { x } & { \text { if } x < - 3 } \\ { x ^ { 2 } } & { \text { if } - 3 \leq x < 3 } \\ { - 6 } & { \text { if } x \geq 3 } \end{array} \right.$
$f ( x ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } \quad x < - 1 } \\ { x ^ { 2 } } & { \text { if } } { - 1 \leq x \leq 0 } \\ { \frac { 1 } { x } } & { \text { if } } { x > 0 } \end{array} \right.$
$g ( x ) = \left\{ \begin{array} { c c c } { 1 } & { \text { if } } & { x \leq - 1 } \\ { 0 } & { \text { if } } & { - 1 < x \leq 1 } \\ { - 1 } & { \text { if } } & { x > 1 } \end{array} \right.$
$g ( x ) = \left[\!\![x]\!\!\right] + 2$

Answer

Exercise $\PageIndex{12}$

Evaluate.

1. $f ( x ) = \left\{ \begin{array} { c l } { 5 x - 2 } & { \text { if } x < - 6 } \\ { x ^ { 2 } } & { \text { if } x \geq - 6 } \end{array} \right.$

Find $f (−10), f (−6)$, and $f (0)$.

2. $h ( x ) = \left\{ \begin{array} { c l } { 2 - 5 x } & { \text { if } x \leq 0 } \\ { x ^ { 3 } } & { \text { if } x > 0 } \end{array} \right.$

Find $h (−1), h (0)$, and $h (\frac{1}{2})$.

3. $g ( x ) = \left\{ \begin{array} { c l } { - 5 } & { \text { if } x < - 4 } \\ { x - 9 } & { \text { if } - 4 \leq x < 0 } \\ { \sqrt { x } } & { \text { if } \quad x \geq 0 } \end{array} \right.$

Find $g (−10), g (0)$ and $g (8)$.

4. $q ( x ) = \left\{ \begin{array} { c c c } { \frac { 1 } { x } } & { \text { if } } & { x < - 1 } \\ { 0 } & { \text { if } } & { - 1 \leq x \leq 1 } \\ { x } & { \text { if } } & { x > 1 } \end{array} \right.$

Find $q (− 5 3 ), q (1)$ and $q (16)$.

Answer

1. $f ( - 10 ) = - 52 , f ( - 6 ) = 36 , f ( 0 ) = 0$

3. $g ( - 10 ) = - 5 , g ( - 4 ) = - 13 , g ( 8 ) = 2 \sqrt { 2 }$

Exercise $\PageIndex{13}$

Sketch the graph of the given function.

$f ( x ) = ( x + 5 ) ^ { 2 } - 10$
$g ( x ) = \sqrt { x - 6 } + 9$
$p ( x ) = x - 9$
$h ( x ) = x ^ { 3 } + 5$
$f ( x ) = | x - 20 | - 40$
$f ( x ) = \frac { 1 } { x - 3 }$
$h ( x ) = \frac { 1 } { x + 3 } - 6$
$g ( x ) = \sqrt [ 3 ] { x - 4 } + 2$
$f ( x ) = \left\{ \begin{array} { l l } { ( x + 4 ) ^ { 2 } } & { \text { if } x < - 2 } \\ { x + 2 } & { \text { if } x \geq - 2 } \end{array} \right.$
$g ( x ) = \left\{ \begin{array} { l l } { - 2 } & { \text { if } x < 6 } \\ { | x - 8 | - 4 } & { \text { if } x \geq 6 } \end{array} \right.$
$g ( x ) = - | x + 4 | - 8$
$h ( x ) = - x ^ { 2 } + 16$
$f ( x ) = \sqrt { - x } - 2$
$r ( x ) = - \frac { 1 } { x } + 2$
$g ( x ) = - 2 | x + 10 | + 8$
$f ( x ) = - 5 \sqrt { x + 1 }$
$f ( x ) = - \frac { 1 } { 4 } x ^ { 2 } + 1$
$h ( x ) = \frac { 1 } { 3 } ( x - 1 ) ^ { 3 } + 2$

Answer

11.

13.

15.

17.

Exercise $\PageIndex{14}$

Write an equation that represents the function whose graph is given.

Answer

1. $f ( x ) = ( x - 4 ) ^ { 2 } - 6$

3. $f ( x ) = - x ^ { 2 } + 4$

5. $f ( x ) = - x ^ { 3 } - 2$

7. $f ( x ) = - 10$

Exercise $\PageIndex{15}$

Solve.

$|5x − 4| = 14$
$|4 − 3x| = 4$
$9 − 5 |x − 4| = 4$
$6 + 2 |x + 10| = 12$
$|3x − 6| + 5 = 5$
$0.2 |x − 1.8| = 4.6$
$\frac { 2 } { 3 } \left| 2 x - \frac { 1 } { 2 } \right| + \frac { 1 } { 3 } = 2$
$\frac { 1 } { 4 } \left| x + \frac { 5 } { 2 } \right| - 2 = \frac { 1 } { 8 }$
$| 3 x - 9 | = | 4 x + 3 |$
$| 9 x - 7 | = | 3 + 8 x |$

Answer

1. $- 2 , \frac { 18 } { 5 }$

3. $3,5$

5. $2$

7. $- 1 , \frac { 3 } { 2 }$

9. $- 12 , \frac { 6 } { 7 }$

Exercise $\PageIndex{16}$

Solve. Graph the solutions on a number line and give the corresponding interval notation.

$| 2 x + 3 | < 1$
$| 10 x - 15 | \leq 25$
$| 6 x - 1 | \leq 11$
$| x - 12 | > 7$
$6 - 4 \left| x - \frac { 1 } { 2 } \right| \leq 2$
$5 - | x + 6 | \geq 4$
$| 3 x + 1 | + 7 \leq 4$
$2 | x - 3 | + 6 > 4$
$5 \left| \frac { 1 } { 3 } x - \frac { 1 } { 2 } \right| > \frac { 5 } { 6 }$
$6.4 - 3.2 | x + 1.6 | > 0$

Answer

1. $( - 2 , - 1 )$;

3. $\left[ - \frac { 5 } { 3 } , 2 \right]$;

5. $\left( - \infty , - \frac { 1 } { 2 } \right] \cup \left[ \frac { 3 } { 2 } , \infty \right)$;

7. $\emptyset$;

9. $( - \infty , 1 ) \cup ( 2 , \infty )$;

Exercise $\PageIndex{17}$

Is the ordered pair a solution to the given inequality?

$9 x - 2 y < - 1 ; ( - 1 , - 3 )$;
$4 x + \frac { 1 } { 3 } y > 0 ; ( 1 , - 12 )$;
$\frac { 3 } { 4 } x - y \geq \frac { 1 } { 2 } ; \left( \frac { 1 } { 2 } , - \frac { 1 } { 4 } \right)$
$x - y \leq - 6 ; ( - 1,7 )$
$y \leq x ^ { 2 } - 3 ; ( - 3,5 )$
$y > | x - 6 | + 10 ; ( - 4,12 )$
$y < ( x - 1 ) ^ { 3 } + 7 ; ( - 1,0 )$
$y \geq \sqrt { x + 4 } ; ( - 3,4 )$

Answer

1. Yes

3. Yes

5. Yes

7. No

Exercise $\PageIndex{18}$

Graph the solution set.

$x + y < 6$
$2x − 3y ≥ 9$
$3x − y ≤ 6$
$y + 4 > 0$
$x − 6 ≥ 0$
$−\frac{1}{3} x +\frac{1}{6} y >\frac{1}{2}$
$y > (x − 2)^{2} − 3$
$y ≤ (x + 6)^{2} + 3$
$y < − |x| + 9$
$y > |x − 12| + 3$
$y ≥ x^{3} + 8$
$y > −(x − 2)^{3}$

Answer

11.

Sample Exam

Exercise $\PageIndex{19}$

1. Determine whether or not the following graph represents a function or not. Explain.

2. Determine the domain and range of the following function.

3. Given $g (x) = x^{2} − 5x + 1$, find $g (−1), g (0)$, and $g (x + h)$.

4. Given the graph of a function $f$:

(a) Find $f(-6), f(0),$ and $f(2)$

(b) Find $x$ where $f(x)=2$

5. Graph $f ( x ) = - \frac { 5 } { 2 } x + 7$ and label the $x$-intercept.

6. Find a linear function passing through $(−\frac{1}{2}, −1)$ and $(2, −2)$.

7. Find the equation of the line parallel to $2x − 6y = 3$ and passing through $(−1, −2)$.

8. Find the equation of the line perpendicular to $3x − 4y = 12$ and passing through $(6, 1)$.

9. The annual revenue of a new web-services company in dollars is given by $R (n) = 125n$, where $n$ represents the number of users the company has registered. The annual maintenance cost of the company’s registered user base in dollars is given by the formula $C (n) = 85n + 22, 480$ where $n$ represents the users.

(a) Write a function that models the annual profit based on the number of registered users.

(b) Determine the number of registered users needed to break even.

10. A particular search engine assigns a ranking to a webpage based on the number of links that direct users to the webpage. If no links are found, the webpage is assigned a ranking of $1$. If $40$ links are found directing users to the webpage, the search engine assigns a page ranking of $5$.

(a) Find a linear function that gives the webpage ranking based on the number of links that direct users to it.

(b) How many links will be needed to obtain a page ranking of $7$?

Answer

1. The graph is not a function; it fails the vertical line test.

3. $g (−1) = 7, g (0) = 1$, and $g (x + h) = x^{2} + 2xh + h^{2} − 5x − 5h + 1$