Question

How do you find the domain and range of `f(x) = (x+7) / (x-5)`?

Answer 1

I got a domain and range of:

`(-oo, 5) uu (5, oo)`, or `x ne 5`

`(-oo, 1) uu (1, oo)`, or `y ne 1`

---

The function is undefined for `x` values when the denominator, `x - 5`, is `0`; it's undefined to divide by `0`. Therefore, when `x = 5`, `f(x)` is undefined.

`f(5) = (5 + 7)/(5 - 5) = color(green)(12/0)`

Since the domain is based on the allowed values of `x`, the domain is:

`color(blue)((-oo,5) uu (5,oo))`

Based on the domain, we would find the range by solving for `x` in terms of `f(x)`, which we will write as `y = f(x)`.

`y = (x + 7)/(x - 5)`

`y(x-5) = x + 7`

`xy - 5y = x + 7`

`x - xy = -5y - 7`

`x(1 - y) = -5y - 7`

`x = (-5y - 7)/(1 - y)`

`color(green)(x = (5y + 7)/(y - 1))`

This means when `y = 1`, the function is undefined as well. So, the range is:

`color(blue)((-oo, 1) uu (1, oo))`

You can see that this is the case in the graph itself:

graph{(x + 7)/(x - 5) [-73.3, 74.9, -37.07, 36.97]}

What you should notice is the horizontal asymptote at `y = 1`, and the vertical asymptote at `x = 5`.

Because the function is trying to reach an undefined value at those points (`x ne 5`, `y ne 1`), you get these "walls" that cannot be crossed, only ascended or descended from either side.