Algebra Examples

$f (x) = \sqrt[3]{2 x + 7}$ , $g (x) = \frac{1}{2} x^{3} - \frac{7}{2}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\sqrt[3]{2 x + 7})$ by substituting in the value of $f$ into $g$ .

$g (\sqrt[3]{2 x + 7}) = \frac{1}{2} \cdot {(\sqrt[3]{2 x + 7})}^{3} - \frac{7}{2}$

Step 3

Simplify each term.

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Step 3.1

Rewrite ${\sqrt[3]{2 x + 7}}^{3}$ as $2 x + 7$ .

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 x + 7}$ as ${(2 x + 7)}^{\frac{1}{3}}$ .

$g (\sqrt[3]{2 x + 7}) = \frac{1}{2} \cdot {({(2 x + 7)}^{\frac{1}{3}})}^{3} - \frac{7}{2}$

Step 3.1.2

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

$g (\sqrt[3]{2 x + 7}) = \frac{1}{2} \cdot {(2 x + 7)}^{\frac{1}{3} \cdot 3} - \frac{7}{2}$

Step 3.1.3

Combine $\frac{1}{3}$ and $3$ .

$g (\sqrt[3]{2 x + 7}) = \frac{1}{2} \cdot {(2 x + 7)}^{\frac{3}{3}} - \frac{7}{2}$

Step 3.1.4

Cancel the common factor of $3$ .

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Step 3.1.4.1

Cancel the common factor.

$g (\sqrt[3]{2 x + 7}) = \frac{1}{2} \cdot {(2 x + 7)}^{\frac{3}{3}} - \frac{7}{2}$

Step 3.1.4.2

Rewrite the expression.

$g (\sqrt[3]{2 x + 7}) = \frac{1}{2} \cdot (2 x + 7) - \frac{7}{2}$

Step 3.1.5

Simplify.

$g (\sqrt[3]{2 x + 7}) = \frac{1}{2} \cdot (2 x + 7) - \frac{7}{2}$

Step 3.2

Apply the distributive property.

$g (\sqrt[3]{2 x + 7}) = \frac{1}{2} \cdot (2 x) + \frac{1}{2} \cdot 7 - \frac{7}{2}$

Step 3.3

Cancel the common factor of $2$ .

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Step 3.3.1

Factor $2$ out of $2 x$ .

$g (\sqrt[3]{2 x + 7}) = \frac{1}{2} \cdot (2 (x)) + \frac{1}{2} \cdot 7 - \frac{7}{2}$

Step 3.3.2

Cancel the common factor.

$g (\sqrt[3]{2 x + 7}) = \frac{1}{2} \cdot (2 x) + \frac{1}{2} \cdot 7 - \frac{7}{2}$

Step 3.3.3

Rewrite the expression.

$g (\sqrt[3]{2 x + 7}) = x + \frac{1}{2} \cdot 7 - \frac{7}{2}$

Step 3.4

Combine $\frac{1}{2}$ and $7$ .

$g (\sqrt[3]{2 x + 7}) = x + \frac{7}{2} - \frac{7}{2}$

Step 4

Combine the opposite terms in $x + \frac{7}{2} - \frac{7}{2}$ .

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Step 4.1

Combine the numerators over the common denominator.

$g (\sqrt[3]{2 x + 7}) = x + \frac{7 - 7}{2}$

Step 4.2

Subtract $7$ from $7$ .

$g (\sqrt[3]{2 x + 7}) = x + \frac{0}{2}$

Step 4.3

Divide $0$ by $2$ .

$g (\sqrt[3]{2 x + 7}) = x + 0$

Step 4.4

Add $x$ and $0$ .

$g (\sqrt[3]{2 x + 7}) = x$