Question

Graph each function. State the domain and range.

$f\left(x\right)=\left\{\begin{array}{l}2x-3\text{if}x\le 2\\ x+1\text{if}x>2\end{array}\right\}$

Short answer

The domain is all real numbers and the range is $\left\{y:y>3\text{or}y\le 1\right\}$.

Step-by-step solution

Step 1. State the concept for the given function.

This is a piece-wise defined function. Check for different values in the domain and include the values for which the function changes :

$y=f\left(x\right)=\left\{\begin{array}{c}2x-3;x\le 2\\ x+1;x>2\end{array}\right\}$

Step 2. Make a table for the given function.

Construct a table for $f\left(x\right)$by taking some values of $x$.

$x$	$x>2$	$x\le 2$	$y=f\left(x\right)=\left\{\begin{array}{c}2x-3;x\le 2\\ x+1;x>2\end{array}\right\}$
0	No	Yes	$-3$
1	No	Yes	$-1$
2	No	Yes	1
3	Yes	No	4
4	Yes	No	5

Step 3. Plot the graph.

Plot the values of $x$and $f\left(x\right)$on a graph,

The $x$-coordinate values specify the domain of the function. Since the graph covers all possible values of $x$, the domain is all real numbers.

The $y$-coordinate values specify the range of the function. Since, the graph does not take any value less than or equal to$y=3$and more than $y=-1$, the range is $\left\{y:y>3\text{or}y\le -1\right\}$.