Algebra Examples

$f (x) = \sqrt{x}$ $g (x) = \frac{1}{\sqrt{x}}$ $h (x) = x^{2}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{1}{\sqrt{x}})$ by substituting in the value of $g$ into $f$ .

$f (\frac{1}{\sqrt{x}}) = \sqrt{\frac{1}{\sqrt{x}}}$

Step 3

Remove parentheses.

$f (\frac{1}{\sqrt{x}}) = \sqrt{\frac{1}{\sqrt{x}}}$

Step 4

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$f (\frac{1}{\sqrt{x}}) = \sqrt{\frac{1}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$f (\frac{1}{\sqrt{x}}) = \sqrt{\frac{\sqrt{x}}{\sqrt{x} \sqrt{x}}}$

Step 5.2

Raise $\sqrt{x}$ to the power of $1$ .

$f (\frac{1}{\sqrt{x}}) = \sqrt{\frac{\sqrt{x}}{\sqrt{x} \sqrt{x}}}$

Step 5.3

Raise $\sqrt{x}$ to the power of $1$ .

$f (\frac{1}{\sqrt{x}}) = \sqrt{\frac{\sqrt{x}}{\sqrt{x} \sqrt{x}}}$

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

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

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

Step 5.6.3

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

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

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

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

Step 5.6.4.2

Rewrite the expression.

$f (\frac{1}{\sqrt{x}}) = \sqrt{\frac{\sqrt{x}}{x}}$

Step 5.6.5

Simplify.

$f (\frac{1}{\sqrt{x}}) = \sqrt{\frac{\sqrt{x}}{x}}$

Step 6

Rewrite $\sqrt{\frac{\sqrt{x}}{x}}$ as $\frac{\sqrt{\sqrt{x}}}{\sqrt{x}}$ .

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt{\sqrt{x}}}{\sqrt{x}}$

Step 7

Rewrite $\sqrt{\sqrt{x}}$ as $\sqrt[4]{x}$ .

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x}}{\sqrt{x}}$

Step 8

Multiply $\frac{\sqrt[4]{x}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{\sqrt[4]{x}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 9.2

Raise $\sqrt{x}$ to the power of $1$ .

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 9.3

Raise $\sqrt{x}$ to the power of $1$ .

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 9.4

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

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 9.5

Add $1$ and $1$ .

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x} \sqrt{x}}{{\sqrt{x}}^{2}}$

Step 9.6

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

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Step 9.6.1

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

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}$

Step 9.6.2

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

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

Step 9.6.3

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

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 9.6.4.2

Rewrite the expression.

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x} \sqrt{x}}{x}$

Step 9.6.5

Simplify.

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x} \sqrt{x}}{x}$

Step 10

Simplify the numerator.

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Step 10.1

Rewrite the expression using the least common index of $4$ .

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Step 10.1.1

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

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x} x^{\frac{1}{2}}}{x}$

Step 10.1.2

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{2}{4}}$ .

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x} x^{\frac{2}{4}}}{x}$

Step 10.1.3

Rewrite $x^{\frac{2}{4}}$ as $\sqrt[4]{x^{2}}$ .

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x} \sqrt[4]{x^{2}}}{x}$

Step 10.2

Combine using the product rule for radicals.

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x \cdot x^{2}}}{x}$

Step 10.3

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

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Step 10.3.1

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

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Step 10.3.1.1

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

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x \cdot x^{2}}}{x}$

Step 10.3.1.2

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

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x^{1 + 2}}}{x}$

Step 10.3.2

Add $1$ and $2$ .

$f (\frac{1}{\sqrt{x}}) = \frac{\sqrt[4]{x^{3}}}{x}$