Question

How do you find domain and range of a rational function?

Answer 1

The domain of a rational function is all real numbers that make the denominator nonzero, which is fairly easy to find; however, the range of a rational function is not as easy to find as the domain. You will have to know the graph of the function to find its range.

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Example 1

`f(x)=x/{x^2-4}`

`x^2-4=(x+2)(x-2) ne 0 Rightarrow x ne pm2`,

So, the domain of `f` is

`(-infty,-2)cup(-2,2)cup(2,infty)`.

The graph of `f(x)` looks like:

Since the middle piece spans from `-infty` to `+infty`, the range is `(-infty,infty)`.

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Example 2

`g(x)={x^2+x}/{x^2-2x-3}`

`x^2-2x-3=(x+1)(x-3) ne 0 Rightarrow x ne -1, 3`

So, the domain of `g` is:

`(-infty,-1)cup(-1,3)cup(3,infty)`.

The graph of `g(x)` looks like this:

Since `g` never takes the values `1/4` or `1`, the range of `g(x)` is
`(-infty,1/4)cup(1/4,1)cup(1,infty)`.

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I hope that this was helpful.