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