Question

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

Answer 1

domain: `{x in RR|x!=3}`
range: `{y in RR|y!=0}`

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 `color(green)(|bar(ul(color(white)(a/a){x in RR|x!=3}color(white)(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 `color(green)(|bar(ul(color(white)(a/a){y in RR|y!=0}color(white)(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!=0`

`x!=3`

Thus, the domain is `color(blue)(|bar(ul(color(white)(a/a){x in RR|x!=3}color(white)(a/a)|)))`.

Recall that `y=1/(x-3)` can be written as `y=1/(x-3)+0`. The `+0` indicates the restriction for the `y` values of the function. It states that `y!=0`.

Thus, the range is `color(blue)(|bar(ul(color(white)(a/a){yin RR|y!=0}color(white)(a/a)|)))`.