# What is the domain and range of y= 1 / (x-3) ?

Mar 17, 2016

domain: $\left\{x \in \mathbb{R} | x \ne 3\right\}$
range: $\left\{y \in \mathbb{R} | y \ne 0\right\}$

#### Explanation:

Method 1
If you graph the function, it would look like:

graph{1/(x-3) [-10, 10, -5, 5]}

By looking at the graph, you can see that $x$ can be any number. However, even though the $x$ values get closer and closer to $3$, it never reaches $3$. Thus, the domain is $\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \left\{x \in \mathbb{R} | x \ne 3\right\} \textcolor{w h i t e}{\frac{a}{a}} |}}}$.

Similarly, you can see that $y$ can be any number as well. However, as the $y$ values approach $0$, they only get closer, but never actually reach $0$. Thus, the range is $\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \left\{y \in \mathbb{R} | y \ne 0\right\} \textcolor{w h i t e}{\frac{a}{a}} |}}}$.

Method 2
To determine the domain, set the denominator of the function to cannot equal $0$ and solve for $x$. The result is the restriction part of the domain.

$x - 3 \ne 0$

$x \ne 3$

Thus, the domain is $\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \left\{x \in \mathbb{R} | x \ne 3\right\} \textcolor{w h i t e}{\frac{a}{a}} |}}}$.

Recall that $y = \frac{1}{x - 3}$ can be written as $y = \frac{1}{x - 3} + 0$. The $+ 0$ indicates the restriction for the $y$ values of the function. It states that $y \ne 0$.

Thus, the range is $\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \left\{y \in \mathbb{R} | y \ne 0\right\} \textcolor{w h i t e}{\frac{a}{a}} |}}}$.