Question

find the rule that describes the piece - wise - defined function. write the rule for the function. select the correct choice below and fill in the answer boxes to complete your choice. (type an exact answer, using radicals as needed. do not factor. type integers or simplified fractions.) a. $f(x)=\begin{cases}square&\text{if }squareleq xleqsquare\square&\text{if }squareleq x

Explanation:

Step1: Identify break - points

The break - points of the piece - wise function from the graph are \(x = 0\).

Step2: Find the equation for \(x\leq0\)

The line for \(x\leq0\) passes through \((- 3,1)\) and \((0,0)\). The slope \(m=\frac{0 - 1}{0+3}=-\frac{1}{3}\). Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(0,0)\), the equation is \(y =-\frac{1}{3}x\).

Step3: Find the equation for \(x>0\)

The line for \(x > 0\) passes through \((0,0)\) and \((8,2)\). The slope \(m=\frac{2 - 0}{8 - 0}=\frac{1}{4}\). Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(0,0)\), the equation is \(y=\frac{1}{4}x\).

The piece - wise function is \(f(x)=

$$\begin{cases}-\frac{1}{3}x&\text{if }x\leq0\\\frac{1}{4}x&\text{if }x > 0\end{cases}$$

\), which corresponds to option C.

Answer:

C. \(f(x)=

$$\begin{cases}-\frac{1}{3}x&\text{if }x\leq0\\\frac{1}{4}x&\text{if }0\leq x<\infty\end{cases}$$

\)