Algebra Examples

$f (x) = 9 x^{2} + 1$ $g (x) = \sqrt{2 x^{3}}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

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

$f (\sqrt{2 x^{3}}) = 9 {(\sqrt{2 x^{3}})}^{2} + 1$

Step 3

Simplify each term.

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Step 3.1

Rewrite $2 x^{3}$ as $x^{2} \cdot (2 x)$ .

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Step 3.1.1

Factor out $x^{2}$ .

$f (\sqrt{2 x^{3}}) = 9 {\sqrt{2 (x^{2} x)}}^{2} + 1$

Step 3.1.2

Reorder $2$ and $x^{2}$ .

$f (\sqrt{2 x^{3}}) = 9 {\sqrt{x^{2} \cdot (2 x)}}^{2} + 1$

Step 3.1.3

Add parentheses.

$f (\sqrt{2 x^{3}}) = 9 {\sqrt{x^{2} \cdot (2 x)}}^{2} + 1$

Step 3.2

Pull terms out from under the radical.

$f (\sqrt{2 x^{3}}) = 9 {(x \sqrt{2 x})}^{2} + 1$

Step 3.3

Apply the product rule to $x \sqrt{2 x}$ .

$f (\sqrt{2 x^{3}}) = 9 (x^{2} {\sqrt{2 x}}^{2}) + 1$

Step 3.4

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

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Step 3.4.1

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

$f (\sqrt{2 x^{3}}) = 9 (x^{2} {({(2 x)}^{\frac{1}{2}})}^{2}) + 1$

Step 3.4.2

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

$f (\sqrt{2 x^{3}}) = 9 (x^{2} {(2 x)}^{\frac{1}{2} \cdot 2}) + 1$

Step 3.4.3

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

$f (\sqrt{2 x^{3}}) = 9 (x^{2} {(2 x)}^{\frac{2}{2}}) + 1$

Step 3.4.4

Cancel the common factor of $2$ .

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Step 3.4.4.1

Cancel the common factor.

$f (\sqrt{2 x^{3}}) = 9 (x^{2} {(2 x)}^{\frac{2}{2}}) + 1$

Step 3.4.4.2

Rewrite the expression.

$f (\sqrt{2 x^{3}}) = 9 (x^{2} (2 x)) + 1$

Step 3.4.5

Simplify.

$f (\sqrt{2 x^{3}}) = 9 (x^{2} (2 x)) + 1$

Step 3.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.5.1

Move $x$ .

$f (\sqrt{2 x^{3}}) = 9 (x \cdot x^{2} \cdot 2) + 1$

Step 3.5.2

Multiply $x$ by $x^{2}$ .

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Step 3.5.2.1

Raise $x$ to the power of $1$ .

$f (\sqrt{2 x^{3}}) = 9 (x \cdot x^{2} \cdot 2) + 1$

Step 3.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\sqrt{2 x^{3}}) = 9 (x^{1 + 2} \cdot 2) + 1$

Step 3.5.3

Add $1$ and $2$ .

$f (\sqrt{2 x^{3}}) = 9 (x^{3} \cdot 2) + 1$

Step 3.6

Move $2$ to the left of $x^{3}$ .

$f (\sqrt{2 x^{3}}) = 9 (2 \cdot x^{3}) + 1$

Step 3.7

Multiply $2$ by $9$ .

$f (\sqrt{2 x^{3}}) = 18 x^{3} + 1$