4.2E: Exercises - Understanding the Basic Functions

Exercise $\PageIndex{1}$

Match the graph to the function definition.

Figure 2.4.17

Figure 2.4.18

Figure 2.4.19

Figure 2.4.20

Figure 2.4.21

Figure 2.4.22

1. $f(x) = x$
2. $f(x) = x^{2}$
3. $f(x) = x^{3}$
4. $f(x) = |x|$
5. $f(x) = \sqrt{x}$
6. $f(x) = \frac{1}{x}$

Answer

1. $b$

3. $c$

5. $a$

Exercise $\PageIndex{2}$

Evaluate.

1. $f(x) = x$; find $f(−10), f(0)$, and $f(a)$.
2. $f(x) = x^{2}$; find $f(−10), f(0)$, and $f(a)$.
3. $f(x) = x^{3}$; find $f(−10), f(0)$, and $f(a)$.
4. $f(x) = |x|$; find $f(−10), f(0)$, and $f(a)$.
5. $f(x) = \sqrt{x}$; find $f(25), f(0)$, and $f(a)$ where $a ≥ 0$.
6. $f(x) = \frac{1}{x}$; find $f(−10), f (\frac{1}{5})$, and $f(a)$ where $a ≠ 0$.
7. $f(x) = 5$; find $f(−10), f(0)$, and $f(a)$.
8. Graph $f(x) = 5$ and state its domain and range.

Answer

1. $f ( - 10 ) = - 10 , f ( 0 ) = 0 , f ( a ) = a$

3. $f ( - 10 ) = - 1,000 , f ( 0 ) = 0 , f ( a ) = a ^ { 3 }$

5. $f ( 25 ) = 5 , f ( 0 ) = 0 , f ( a ) = \sqrt { a }$

7. $f ( - 10 ) = 5 , f ( 0 ) = 5 , f ( a ) = 5$

8. Domain: $(- \infty , \infty)$; range $\{5\}$

Figure 2.4.23

Exercise $\PageIndex{3}$

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

1.

Figure 2.4.25

2.

Figure 2.4.26

3.

Figure 2.4.27

4.

Figure 2.4.28

Answer

1. $\left( \frac { 3 } { 2 } , \frac { 27 } { 8 } \right)$

3. $\left( - \frac { 5 } { 2 } , - \frac { 5 } { 2 } \right)$

Exercise $\PageIndex{4}$

Evaluate.

1. $f ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } } & { \text { if } x \leq 0 } \\ { x + 2 } & { \text { if } x > 0 } \end{array} \right.$

Find $f(-5), f(0)$, and $f(3)$.

2. $g ( x ) = \left\{ \begin{array} { l } { x ^ { 3 } \text { if } x \leq - 2 } \\ { | x | \text { if } x > - 2 } \end{array} \right.$

Find $g(−3), g(−2)$, and $g(−1)$.

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

Find $h(−2), h(0)$, and $h(4)$.

4. $h ( x ) = \left\{ \begin{array} { l } { - 3 x \text { if } x \leq 0 } \\ { x ^ { 3 } \text { if } 0 < x \leq 4 } \\ { \sqrt { x } \text { if } x > 4 } \end{array} \right.$

Find $h(−5), h(4)$, and $h(25)$.

Answer

1. $f (−5) = 25, f(0) = 0$, and $f(3) = 5$

3. $h(−2) = −5, h(0) = −3$, and $h(4) = 16$

Exercise $\PageIndex{5}$

Graph the piecewise functions.

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

Answer

1.

Figure 2.4.29

3.

Figure 2.4.50

5.

Figure 2.4.51

7.

Figure 2.4.52

9.

Figure 2.4.53

11.

Figure 2.4.54

Exercise $\PageIndex{6}$

Evaluate given the graph of $f$.

1. Find $f(-4), f(-2)$, and $f(0)$.

Figure 2.4.59

2. Find $f(−3), f(0)$, and $f(1)$.

Figure 2.4.60

3. Find $f(0), f(2)$, and $f(4)$.

Figure 2.4.61

4. Find $f(−5), f(−2)$, and $f(2)$.

Figure 2.4.62

5. Find $f(−3), f(−2)$, and $f(2)$.

Figure 2.4.63

6. Find $f(−3), f(0)$, and $f(4)$.

Figure 2.4.64

7. The value of an automobile in dollars is given in terms of the number of years since it was purchased new in $1975$:

Figure 2.4.67

(1) Determine the value of the automobile in the year $1980$.

(2) In what year is the automobile valued at $$9,000$?

8. The cost per unit in dollars of custom lamps depends on the number of units produced according to the following graph:

Figure 2.4.68

(1) What is the cost per unit if $250$ custom lamps are produced?

(2) What level of production minimizes the cost per unit?

9. An automobile salesperson earns a commission based on total sales each month $x$ according to the function:

$( x ) = \left\{ \begin{array} { l l } { 0.03 x\quad \text { if } \quad 0 \leq x < \$ 20,000 } \\ { 0.05 x \quad\text { if } } \quad { \$ 20,000 \leq x < \$ 50,000 } \\ { 0.07 x \quad\text { if } }\quad { x \geq \$ 50,000 } \end{array} \right.$

(1) If the salesperson’s total sales for the month are $$35,500$, what is her commission according to the function?

(2) To reach the next level in the commission structure, how much more in sales will she need?

10. A rental boat costs $$32$ for one hour, and each additional hour or partial hour costs $$8$. Graph the cost of the rental boat and determine the cost to rent the boat for $4 \frac{1}{2}$ hours.

Answer

1. $f(−4) = 1, f(−2) = 1$, and $f(0) = 0$

3. $f(0) = 0, f(2) = 8$, and $f(4) = 0$

5. $f(−3) = 5, f(−2) = 4$, and $f(2) = 2$

7. (1) $$3,000$; (2) $2005$

9. (1) $$1,775$; (2) $$14,500$