# Thread: Some clarification on ranges when combining functions

1. ## Some clarification on ranges when combining functions

A question I just tried has me thinking about the range of a composite function:

$
\begin{array}{lll}
f: x\to x-\frac{1}{x} & [1, \infty) & [0, \infty) \\
g: x\to 3x^2 + 2 & [0, \infty) & [2, \infty) \\
\end{array}
$

Find $
gf(x) = \frac{3}{x^2}+7
$

$
\begin{array}{l}
3(x-\frac{1}{x})^2 + 2 = \frac{3}{x^2} + 8 \\
3x^2 = 12 \\
x = \pm 2
\end{array}
$

The range of g is $[2,\infty)$ though, so is the only answer $x = 2$ or is $x=-2$ also an answer?

Thanks

2. Since you restricted the domain of f, it is not permitted to substitute -2 in for x in the function f. So x=-2 cannot be a solution to the given equation.

3. Originally Posted by DrSteve
Since you restricted the domain of f, it is not permitted to substitute -2 in for x in the function f. So x=-2 cannot be a solution to the given equation.
Ah thank you, I was thinking only in terms of the range and not the domain.