# 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

*) can take, as a consequence of the function's domain.*

**y**## Answer and Explanation:

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

*. It is defined for all set of real numbers except 1. The*

**x = 1****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.

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from Math 105: Precalculus Algebra

Chapter 4 / Lesson 9