Question

Sketch the graph of the piece wise defined function $f\left(x\right)=\left\{\begin{array}{l}2x-x^2\text{if}x>1\\ {\left(x-1\right)}^3\text{if}x\le 1\end{array}\right\}$.

Short answer

The graph of the piece wise defined function $f\left(x\right)=\left\{\begin{array}{l}2x-x^2\text{if}x>1\\ {\left(x-1\right)}^3\text{if}x\le 1\end{array}\right\}$is shown.

Step-by-step solution

Step 1. Construct the table for the given function.

The given function is $f\left(x\right)=\left\{\begin{array}{l}2x-x^2\text{if}x>1\\ {\left(x-1\right)}^3\text{if}x\le 1\end{array}\right\}$whose tables for respective values of $\left(x\right)$and $f\left(x\right)$is shown below.

$x$	2	3	1	0	-1
$f\left(x\right)$	0	-3	0	-1	-8

Step 2. Plot the points on coordinate plane.

By plotting points on coordinate plane, the graph for the piece wise function $f\left(x\right)=\left\{\begin{array}{l}2x-x^2\text{if}x>1\\ {\left(x-1\right)}^3\text{if}x\le 1\end{array}\right\}$is shown below.

Step 3. Description of steps.

The obtained graph is a piece wise graph of the function $f\left(x\right)=\left\{\begin{array}{l}2x-x^2\text{if}x>1\\ {\left(x-1\right)}^3\text{if}x\le 1\end{array}\right\}$.