# 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:

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

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

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

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