1.5E: Piecewise-Defined Functions (Exercises)

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Section 1.5 Exercises

1. Let $ f(x)=\left\{\begin{array}{ccc} {x+1} & {if} & {x<2} \\ {5} & {if} & {x\ge 2} \end{array}\right. $

a) Evaluate $f(1)$, $f(2)$, $f(3)$, and $f(4)$

b) Sketch a graph of the piecewise defined function.

2. Let $ f(x=\left\{\begin{array}{ccc} {4} & {if} & {x<0} \\ {\sqrt{x} } & {if} & {x\ge 0} \end{array}\right.$

a) Evaluate $f(-1)$, $f(0)$, $f(1)$, and $f(2)$

b) Sketch a graph of the piecewise defined function.

3. Let $ f(x)=\left\{\begin{array}{ccc} {3x-1 } & {if} & {x<1} \\ {x+2} & {if} & {x\ge 1} \end{array}\right. $

a) Evaluate $f(0)$, $f(1)$, $f(2)$, and $f(3)$

b) Sketch a graph of the piecewise defined function.

4. Let $ f(x)=\left\{\begin{array}{ccc} {x^2} & {if} & {x<1} \\ {-2x+5 } & {if} & {x\ge 1} \end{array}\right.$

a) Evaluate $f(0)$, $f(1)$, $f(2)$, and $f(3)$

b) Sketch a graph of the piecewise defined function.

5. Sketch a graph of the piecewise defined function $ f(x)=\left\{\begin{array}{ccc} {3} & {if} & {x\le -2} \\ {-x+1} & {if} & {-2<x\le 1} \\ {x+2} & {if} & {x>1} \end{array}\right.$. Then identify the domain and range of $f$.

6. Sketch a graph of the piecewise defined function $ f(x)=\left\{\begin{array}{ccc} {2x+1} & {if} & {x\le 0} \\ {x+1} & {if} & {0<x\le 2} \\ {3} & {if} & {x>2} \end{array}\right.$ Then identify the domain and range of $f$.

7. Write a formula for the piecewise function graphed below.

a) $A piecewise defined function where y = 2 on [-6,-1], y = -2 on (-1,2], and y = -4 on (2,4]$ b) $A piecewise defined function where y = -4 on [-6,-2], y = 5 on (-2,1], and y = -3 on (1,5]$

8. Write a formula for the piecewise function graphed below.

a) $A piecewise defined function where y = 3 below x = 0 and upward opening parabola above x =0$ b) $A piecewise defined function where y is a downward sloping line below x = 0, is an upward sloping line on [0,2], and y = 3 above x = 2$

9. Write a formula for the piecewise function graphed below.

a) $A piecewise defined function where one piece is a line that increases from (-3,-3) to (-1,1), the next is a line from (-1,-2) to (2,1), and the third piece is a line from (2,-2) to (5,-2).$ b) $A piecewise defined function where one piece is a line from (-6,1) to (-2,-1), the next is a line from (-2,-2) to (1,-2), and the third piece is a line from (1,-3) to (6,4.5).$

10. Suppose a commercial cloud data plan charges $5 per year plus $2.5 per GB for the first 10 GB and then $1.5 per GB for each additional GB of data. Find a piecewise defined formula for the yearly cost $C$ in terms of the data load $D$ in GB.

11. Suppose a painting company charges $2.40 per $\text{ft}^2$ to paint a house of up to 2000 $\text{ft}^2$. For houses with more than 2000 $\text{ft}^2$ to paint, they charge $2.10 per $\text{ft}^2$ for each additional square foot of area above 2000 $\text{ft}^2$. Find a piecewise defined formula for the yearly cost $C$ in terms of the square footage painted $A$.

12. Suppose an author of a new book receives $30,000 to write it. For the first 50,000 copies sold, they will make $1.24 per copy sold. After that, they will make $1.02 per copy sold. Find a piecewise defined formula for the amount made $A$ in terms of the number of book copies sold $b$.

Answer

1. a) $f(1)=2$, $f(2)=5$, $f(3)=5$, and $f(4)=5$

b) $the graph of a piecewise defined function$

3. a) $f(0)=-1$ $f(1)=3$ $f(2)=4$ $f(3)=4$

b) $the graph of a piecewise defined function$

5. $the graph of a piecewise defined function$

7. a) $f(x) = \begin{cases} 2 & if & -6 \le x \le -1 \\ -2 & if & -1 < x \le 2 \\ -4 & if & 2 < x \le 4 \end{cases}$

b) $f(x) = \begin{cases} -4 & if & -6 \le x \le -2 \\ 5 & if & -2 < x \le 1 \\ -3 & if & 1 < x \le 5 \end{cases}$

9. a) $f(x) = \begin{cases} 2x + 3 & if & -3 \le x < -1 \\ x - 1 & if & -1\le x \le 2 \\ -2 & if & 2 < x \le 5 \end{cases}$

11. $C =f(A) = \begin{cases} 2.40A & if & 0 \le A \le 2000 \\ 2.10A+600 & if & 2000 < A \end{cases}$

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