# Find the domain and range of the function. f(x) = \frac{1}{(x - 1)^2} - 5

## Question:

Find the domain and range of the function.

{eq}f(x) = \displaystyle \frac{1}{\left(x - 1\right)^2} - 5 {/eq}

## Domain and range of rational functions

The domain of a rational function consists the set of all points (usually independent variable, x) for which the function is defined. Range on the other hand, refers to all possible values that the dependent variable (usually denoted as y) can take, as a consequence of the function's domain.

We are to find the domain and range of the function, {eq}f(x) = \frac{1}{(x-1)^2} - 5 {/eq} . Notice that the independent variable here is x. Taking a look at {eq}\frac{1}{(x-1)^2} {/eq}, we can see that it will be undefined if the denominator equals zero. The value of x that will give the rational expression an undefined result is only for x = 1. It is defined for all set of real numbers except 1. The domain therefore of the function is {eq}(-\infty , 1) \cup (1, \infty) , [x\ \epsilon\ \mathbb{R}\ |\ x\ \neq 1] {/eq}.

The range on the other hand refers to which values of the dependent variable y, are defined based on the domain of the function. As a consequence, the range is then {eq}(-5, \infty), [y\ \epsilon\ \mathbb{R}\ |\ y\ > -5] {/eq}. In other words, the function is asymptotic at x=1 and y = -5. 