Precalculus Examples

$f (x) = \frac{1}{x^{2}} - 2$ , $g (x) = \frac{1}{x}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\frac{1}{x^{2}} - 2)$ by substituting in the value of $f$ into $g$ .

$g (\frac{1}{x^{2}} - 2) = \frac{1}{\frac{1}{x^{2}} - 2}$

Step 3

Simplify the denominator.

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Step 3.1

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$g (\frac{1}{x^{2}} - 2) = \frac{1}{\frac{1}{x^{2}} - 2 \cdot \frac{x^{2}}{x^{2}}}$

Step 3.2

Combine $- 2$ and $\frac{x^{2}}{x^{2}}$ .

$g (\frac{1}{x^{2}} - 2) = \frac{1}{\frac{1}{x^{2}} + \frac{- 2 x^{2}}{x^{2}}}$

Step 3.3

Combine the numerators over the common denominator.

$g (\frac{1}{x^{2}} - 2) = \frac{1}{\frac{1 - 2 x^{2}}{x^{2}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$g (\frac{1}{x^{2}} - 2) = 1 (\frac{x^{2}}{1 - 2 x^{2}})$

Step 5

Multiply $\frac{x^{2}}{1 - 2 x^{2}}$ by $1$ .

$g (\frac{1}{x^{2}} - 2) = \frac{x^{2}}{1 - 2 x^{2}}$