Q. 44 - 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, 

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

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,

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