# Some clarification on ranges when combining functions

• Jan 18th 2011, 09:28 AM
alexgeek
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
• Jan 18th 2011, 09:46 AM
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.
• Jan 18th 2011, 09:52 AM
alexgeek
Quote:

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.