Find g(f(x)) f(x)=5x^2-5 , g(x)=4/x

$f (x) = 5 x^{2} - 5$ , $g (x) = \frac{4}{x}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (5 x^{2} - 5)$ by substituting in the value of $f$ into $g$ .

$g (5 x^{2} - 5) = \frac{4}{5 x^{2} - 5}$

Step 3

Simplify the denominator.

Factor $5$ out of $5 x^{2} - 5$ .

Factor $5$ out of $5 x^{2}$ .

$g (5 x^{2} - 5) = \frac{4}{5 (x^{2}) - 5}$

Factor $5$ out of $- 5$ .

$g (5 x^{2} - 5) = \frac{4}{5 (x^{2}) + 5 (- 1)}$

Factor $5$ out of $5 (x^{2}) + 5 (- 1)$ .

$g (5 x^{2} - 5) = \frac{4}{5 (x^{2} - 1)}$

Rewrite $1$ as $1^{2}$ .

$g (5 x^{2} - 5) = \frac{4}{5 (x^{2} - 1^{2})}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$g (5 x^{2} - 5) = \frac{4}{5 (x + 1) (x - 1)}$