First slide

Functions (XII)

Question

If $f (x) and g (x)$ are two functions with $g (x) = x - \frac{1}{x} and f o g (x) = x^{3} - \frac{1}{x^{3}},$ then $f (x) =$

Moderate

A

$x^{3} + 3 x$
B

$x^{2} - \frac{1}{x^{2}}$
C

$1 + \frac{1}{x^{2}}$
D

$3 x^{2} + \frac{3}{x^{4}}$

Solution

$\begin{array}{l} f o g (x) = x^{3} - \frac{1}{x^{3}}, \\ f (g (x)) = x^{3} - \frac{1}{x^{3}} = {(x - \frac{1}{x})}^{3} + 3 (x - \frac{1}{x}) \end{array}$
$\begin{array}{l} f (x - \frac{1}{x}) = {(x - \frac{1}{x})}^{3} + 3 (x - \frac{1}{x}) \\ let x - \frac{1}{x} = t, f (t) = t^{3} + 3 t \\ Thus f (x) = x^{3} + 3 x \end{array}$

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