Question

Graph each function. State the domain and range.

$\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\left\{\begin{array}{l}x-2\text{if}x<1\\ 3x\text{if}x\ge 1\end{array}\right\}$

Short answer

The domain is all real numbers and the range is

$\mathbf{\{}\mathbf{y}\mathbf{:}\mathbf{y}\mathbf{\ge }\mathbf{3}\mathbf{\text{or}}\mathbf{y}\mathbf{<}\mathbf{-}\mathbf{1}\mathbf{\}}$

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 :

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

Step 2. Make a table for the given function.

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

$x$	$x\ge 1$	$x<1$	$y=f\left(x\right)=\left\{\begin{array}{c}x-2;x<1\\ 3x;x\ge 1\end{array}\right\}$
$-1$	No	Yes	$-3$
0	No	Yes	$-2$
1	Yes	No	3
2	Yes	No	6
3	Yes	No	9

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 $y$-coordiante values specify the range of the function. Since, the graph does not take any value less than$y=3$ and more than or equal to $y=-1$, the range is$\{y:y\ge 3\text{or}y<-1\}$.