Algebra Examples

$f (x) = x^{2}$ ; $g (x) = \sqrt{625 - x^{2}}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

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

$g (x^{2}) = \sqrt{625 - {(x^{2})}^{2}}$

Step 3

Rewrite $625$ as $25^{2}$ .

$g (x^{2}) = \sqrt{25^{2} - {(x^{2})}^{2}}$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 25$ and $b = x^{2}$ .

$g (x^{2}) = \sqrt{(25 + x^{2}) (25 - x^{2})}$

Step 5

Simplify.

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Step 5.1

Rewrite $25$ as $5^{2}$ .

$g (x^{2}) = \sqrt{(25 + x^{2}) (5^{2} - x^{2})}$

Step 5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

$g (x^{2}) = \sqrt{(25 + x^{2}) (5 + x) (5 - x)}$