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

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

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Answer 1 · 2014-10-30T03:48:18

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.

Example 1

$f (x) = \frac{x}{x^{2} - 4}$ $f(x)=x/{x^2-4}$

$x^{2} - 4 = (x + 2) (x - 2) \neq 0 \Rightarrow x \neq \pm 2$ $x^2-4=(x+2)(x-2) ne 0 Rightarrow x ne pm2$ ,

So, the domain of $f$ $f$ is

$(- \infty, - 2) \cup (- 2, 2) \cup (2, \infty)$ $(-infty,-2)cup(-2,2)cup(2,infty)$ .

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

enter image source here

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

Example 2

$g (x) = \frac{x^{2} + x}{x^{2} - 2 x - 3}$ $g(x)={x^2+x}/{x^2-2x-3}$

$x^{2} - 2 x - 3 = (x + 1) (x - 3) \neq 0 \Rightarrow x \neq - 1, 3$ $x^2-2x-3=(x+1)(x-3) ne 0 Rightarrow x ne -1, 3$

So, the domain of $g$ $g$ is:

$(- \infty, - 1) \cup (- 1, 3) \cup (3, \infty)$ $(-infty,-1)cup(-1,3)cup(3,infty)$ .

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

enter image source here

Since $g$ $g$ never takes the values $\frac{1}{4}$ $1/4$ or $1$ $1$ , the range of $g (x)$ $g(x)$ is
$(- \infty, \frac{1}{4}) \cup (\frac{1}{4}, 1) \cup (1, \infty)$ $(-infty,1/4)cup(1/4,1)cup(1,infty)$ .

I hope that this was helpful.

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

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