Q. 43 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 41–44, the graph of a piecewise-defined function is given. Write a definition for each function. 

Step-by-Step Solution

Answer

The piecewise-defined function is, 

$f\left(x\right)=\left\{\begin{array}{c}-x for x\le 0\\ -x+2 for 0<x\le 2 \end{array}\right.$

Step 1. Given Information

We are given a graph of a piecewise-defined function, 

Step 2. Defining the function

It can be seen from the graph that the point $(0,0)$ on the graph.

So, when $x=0$, $f\left(0\right)=0$.

Similarly, $(-1,1)$ is a point on the graph means that the y-coordinate $1$ is the value of $f$ at the $x$ coordinate $-1$.

So, when $x=-1$,

$f(-1)=1$

Step 3. Defining the function

It can  be seen from the graph that for the points with the x-coordinates is between $0$ and $-1$, both inclusive, the value of the y- coordinate is opposite of the value of the x-coordinate.

So,

 $$\begin{gathered} f(-1)=-(-1) \\ =1 \end{gathered}$$.

So, the value of the function $f\left(x\right)$, when $-1\le x\le 0$ is given by $-x$.

Step 4. Defining the function

It can be seen from the graph that for the points with the x-coordinates between $0$ and $2$ , with $2$ inclusive, the value of the y-coordinate is $-x+2$

$$\begin{gathered} f\left(1\right)=-1+2 \\ =1 \end{gathered}$$

So, the value of the function when $0<x\le 2$ is given by $-x+2$.

So, the function will be, 

$f\left(x\right)=\left\{\begin{array}{c}-x for x\le 0\\ -x+2 for 0<x\le 2 \end{array}\right.$