# What is the domain and range of f(x) =x^2/ (x^2-6)?

Apr 15, 2018

Domain: x≠sqrt6

Range: $y \in \mathbb{R}$

#### Explanation:

First, it is important to understand the distinction between domain and range.

Domain:
All possible $x$-values for the expression

Range:
All possible $y$-values for the expression

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Finding the Domain:

On the numerator, the $x$-value can be any real number

The denominator however, ≠ 0 since that would make the function undefined.

So, to find the number that $x$ cannot equal in the function, we must write:

Denominator$= 0$

${x}^{2} - 6 = 0$

${x}^{2} = 6$

$\sqrt{{x}^{2}} = \sqrt{6}$

$\therefore x = \sqrt{6}$ when the function is undefined

This means that the domain is: x≠sqrt6

Finding the Range:

$f \left(x\right)$ just means that $x$ is the input of the function.

The actual result of the equation is $y$, do the function can be rewritten as:

$y = {x}^{2} / \left({x}^{2} - 6\right)$

Now there is a $y$-value to work with. Since there are no limitations on the value of $y$ in the equation, it can be any real number.

This means that the range is: $y \in \mathbb{R}$

or

$y$ can be any real number