Calculus Examples

$f (x) = \sqrt{x + 5}$ , $g (x) = 12 x^{2} - 9$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\sqrt{x + 5})$ by substituting in the value of $f$ into $g$ .

$g (\sqrt{x + 5}) = 12 {(\sqrt{x + 5})}^{2} - 9$

Step 3

Simplify each term.

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Step 3.1

Rewrite ${\sqrt{x + 5}}^{2}$ as $x + 5$ .

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 5}$ as ${(x + 5)}^{\frac{1}{2}}$ .

$g (\sqrt{x + 5}) = 12 {({(x + 5)}^{\frac{1}{2}})}^{2} - 9$

Step 3.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$g (\sqrt{x + 5}) = 12 {(x + 5)}^{\frac{1}{2} \cdot 2} - 9$

Step 3.1.3

Combine $\frac{1}{2}$ and $2$ .

$g (\sqrt{x + 5}) = 12 {(x + 5)}^{\frac{2}{2}} - 9$

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

Cancel the common factor.

$g (\sqrt{x + 5}) = 12 {(x + 5)}^{\frac{2}{2}} - 9$

Step 3.1.4.2

Rewrite the expression.

$g (\sqrt{x + 5}) = 12 (x + 5) - 9$

Step 3.1.5

Simplify.

$g (\sqrt{x + 5}) = 12 (x + 5) - 9$

Step 3.2

Apply the distributive property.

$g (\sqrt{x + 5}) = 12 x + 12 \cdot 5 - 9$

Step 3.3

Multiply $12$ by $5$ .

$g (\sqrt{x + 5}) = 12 x + 60 - 9$

Step 4

Subtract $9$ from $60$ .

$g (\sqrt{x + 5}) = 12 x + 51$