Precalculus Examples

$f (x) = \sqrt{49 - x^{2}}$ , $g (x) = \sqrt{x^{2} - 4}$ , $(f - g) (x)$

Step 1

Evaluate $f - g$ .

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Step 1.1

Replace the function designators in $f - g$ with the actual functions.

$(\sqrt{49 - x^{2}}) - (\sqrt{x^{2} - 4})$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite $49$ as $7^{2}$ .

$\sqrt{7^{2} - x^{2}} - (\sqrt{x^{2} - 4})$

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 7$ and $b = x$ .

$\sqrt{(7 + x) (7 - x)} - (\sqrt{x^{2} - 4})$

Step 1.2.3

Rewrite $4$ as $2^{2}$ .

$\sqrt{(7 + x) (7 - x)} - \sqrt{x^{2} - 2^{2}}$

Step 1.2.4

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 $b = 2$ .

$\sqrt{(7 + x) (7 - x)} - \sqrt{(x + 2) (x - 2)}$

Step 1.3

Simplify with commuting.

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Step 1.3.1

Reorder $7$ and $x$ .

$\sqrt{(x + 7) (7 - x)} - \sqrt{(x + 2) (x - 2)}$

Step 1.3.2

Reorder $7$ and $- x$ .

$\sqrt{(x + 7) (- x + 7)} - \sqrt{(x + 2) (x - 2)}$

Step 2

Evaluate $f - g$ at $x$ .

$\sqrt{((x) + 7) (- (x) + 7)} - \sqrt{((x) + 2) ((x) - 2)}$

Step 3

Simplify $\sqrt{((x) + 7) (- (x) + 7)} - \sqrt{((x) + 2) ((x) - 2)}$ .

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Step 3.1

Remove parentheses.

$\sqrt{((x) + 7) (- (x) + 7)} - \sqrt{((x) + 2) ((x) - 2)}$

Step 3.2

Remove parentheses.

$\sqrt{(x + 7) (- x + 7)} - \sqrt{(x + 2) (x - 2)}$