Question

4.7 piecewise functions (pp. 217 - 224)
graph ( y=left{\begin{array}{ll}\frac{3}{2} x+3, & \text { if } x leq 0 \\ -2 x, & \text { if } x>0end{array}
ight. ). describe the domain and range.
step 1 graph ( y=\frac{3}{2} x+3 ) for ( x leq 0 ). because ( x ) is less than or equal to 0, use a closed circle at ( (0,3) ).
step 2 graph ( y=-2 x ) for ( x>0 ). because ( x ) is not equal to 0, use an open circle at ( (0,0) ).
the domain is all real numbers. the range is ( y leq 3 ).

1. evaluate the function in the example when (a) ( x = 0 ) and (b) ( x = 5 ).

graph the function. describe the domain and range.

1. ( y=left{\begin{array}{ll}x+6, & \text { if } x leq 0 \\ -3 x, & \text { if } x>0end{array}

ight. ) 20. ( y=left{\begin{array}{ll}4 x+2, & \text { if } x<-4 \\ 2 x-6, & \text { if } x geq-4end{array}
ight. )
write the absolute value function as a piecewise function.

1. ( y=|x|+15 ) 22. ( y=4|x + 5| ) 23. ( y=2|x + 2|-3 )
2. you are organizing a school fair and rent a popcorn machine for 3 days. the rental company charges ( $ 65 ) for the first day and ( $ 35 ) for each additional day. write and graph a step function that represents the relationship between the number ( x ) of days and the total cost ( y ) (in dollars) of renting the popcorn machine.

Explanation:

Step1: Evaluate for \(x = 0\)

Since \(x=0\) and the piece - wise function is \(y=

$$\begin{cases}\frac{3}{2}x + 3, &\text{if }x\leq0\\- 2x, &\text{if }x > 0\end{cases}$$

\), we use the first part \(y=\frac{3}{2}x+3\). Substitute \(x = 0\) into \(y=\frac{3}{2}x + 3\):
\(y=\frac{3}{2}(0)+3\)
\(y = 3\)

Step2: Evaluate for \(x = 5\)

Since \(x = 5>0\), we use the second part \(y=-2x\). Substitute \(x = 5\) into \(y=-2x\):
\(y=-2\times5\)
\(y=-10\)

Answer:

(a) When \(x = 0\), \(y = 3\). (b) When \(x = 5\), \(y=-10\).