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

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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.

1. $$h (x) = 1 2 x − 3; h (−8), h (3),$$ and $$h (4a + 1)$$
2. $$p (x) = 4 − x ; p (−10), p (0),$$ and $$p (5a − 1)$$
3. $$f (x) = 2x^{2} − x + 3$$; find $$f (−5), f (0)$$, and $$f (x + h)$$
4. $$g (x) = x^{2} − 9$$; find $$f (−3), f (2)$$, and $$f (x + h)$$
5. $$g (x) = \sqrt{2x − 1}$$; find $$g (5), g (1), g (13)$$
6. $$h ( x ) = \sqrt [ 3 ] { x + 6 }$$; find $$h (−7), h (−6)$$, and $$h (21)$$
7. $$f (x) = 8x + 3$$; find $$x$$ where $$f (x) = 10$$.
8. $$g (x) = 5 − 3x$$; find $$x$$ where $$g (x) = −4$$.
9. 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 $$g (x) = 12$$.

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.

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

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Exercise $$\PageIndex{4}$$

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

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

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Exercise $$\PageIndex{5}$$

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

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

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.

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

1. $$x=2$$

3. $$( - 1 , \infty )$$

Exercise $$\PageIndex{7}$$

Find the linear function passing through the given points.

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

6. Find the equation of the given linear function:

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:

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

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.

1. 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 $$9.70$$.
2. 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 $$4,824$$.
3. 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 $$2000$$.
4. 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.
5. 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.
6. 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?
7. 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 $$10,000$$.
8. Determine the breakeven point from the previous exercise.

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 $$P$$.

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

3. $$( - 25,25 )$$

Exercise $$\PageIndex{11}$$

Graph the piecewise defined functions.

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

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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 $$q (16)$$.

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.

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

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Exercise $$\PageIndex{14}$$

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

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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.

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

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.

1. $$| 2 x + 3 | < 1$$
2. $$| 10 x - 15 | \leq 25$$
3. $$| 6 x - 1 | \leq 11$$
4. $$| x - 12 | > 7$$
5. $$6 - 4 \left| x - \frac { 1 } { 2 } \right| \leq 2$$
6. $$5 - | x + 6 | \geq 4$$
7. $$| 3 x + 1 | + 7 \leq 4$$
8. $$2 | x - 3 | + 6 > 4$$
9. $$5 \left| \frac { 1 } { 3 } x - \frac { 1 } { 2 } \right| > \frac { 5 } { 6 }$$
10. $$6.4 - 3.2 | x + 1.6 | > 0$$

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?

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

1. Yes

3. Yes

5. Yes

7. No

Exercise $$\PageIndex{18}$$

Graph the solution set.

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

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## 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 $$7$$?

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

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

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7. $$y = \frac { 1 } { 3 } x - \frac { 5 } { 3 }$$

9. (a) $$P ( n ) = 40 n - 22,480$$ (b) $$562$$ users

Exercise $$\PageIndex{20}$$

Use the transformations to sketch the graph of the following functions and state the domain and range.

1. $$g ( x ) = | x + 4 | - 5$$
2. $$h ( x ) = \sqrt { x - 4 } + 1$$
3. $$r ( x ) = - ( x + 3 ) ^ { 3 }$$
4. Given the graph, determine the function definition and its domain and range:

5. Sketch the graph: h ( x ) = \left\{ \begin{aligned} - x & \text { if } x < 1 \\ \frac { 1 } { x } & \text { if } x \geq 1 \end{aligned} \right.

6. Sketch the graph: $$g ( x ) = - \frac { 1 } { 3 } x ^ { 2 } + 9$$

1. Domain: $$(−∞, ∞)$$; range: $$[−5, ∞)$$

3. Domain: $$(−∞, ∞)$$; range: $$(−∞, ∞)$$

5.

Exercise $$\PageIndex{21}$$

Solve.

1. $$|2 x - 1| + 2 = 7$$
2. $$10 - 5 | 2 x - 3 | = 0$$
3. $$| 7 x + 4 | = | 9 x - 1 |$$

1. $$- 2,3$$

3. $$- \frac { 3 } { 16 } , \frac { 5 } { 2 }$$

Exercise $$\PageIndex{22}$$

Solve and graph the solution set.

1. $$| 2 x - 4 | - 5 < 7$$
2. $$6 + | 3 x - 5 | \geq 13$$
3. $$5 - 3 | x - 4 | \geq - 10$$
4. $$3 | 7 x - 1 | + 5 \leq 2$$

1. $$\left( - \infty , - \frac { 2 } { 3 } \right] \cup [ 4 , \infty )$$;

3. $$\emptyset$$

Exercise $$\PageIndex{23}$$

Graph the solution set.

1. $$\frac { 1 } { 2 } x - \frac { 2 } { 3 } y \geq 4$$
2. $$y > - ( x - 2 ) ^ { 2 } + 4$$