Question

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

Short answer

The piecewise-defined function is,

$f\left(x\right)=\left\{\begin{array}{l}2x+2-1\le x\le 0\\ xx>0\end{array}\right.$

Step-by-step solution

Step 1. Given Information 

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

Step 2. Defining the function  

For the point $\left(0,2\right)$means when $x=0$,

$f\left(2\right)=2$ that means the y-intercept is$2$.

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

So, for $-1\le x\le 0$the function will be,

$f\left(x\right)=2x+2$

Step 3. Defining the function

Similarly, the point $(1,1)$means when $x=1$

$f\left(1\right)=1$and

when $x=2$,

$f\left(2\right)=2$

The line passes to origin so there is no y-intercept.

So, it can be seen from the graph that for $x>0$, the function will be,

$f\left(x\right)=0$

So, the piecewise-defined function will be,

$f\left(x\right)=\left\{\begin{array}{l}2x+2-1\le x\le 0\\ xx>0\end{array}\right.$