Algebra Examples

$f (x) = \sqrt{x + 5}$ , $g (x) = \frac{1}{x}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

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

$f (\frac{1}{x}) = \sqrt{(\frac{1}{x}) + 5}$

Step 3

To write $5$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$f (\frac{1}{x}) = \sqrt{\frac{1}{x} + \frac{5 x}{x}}$

Step 4

Combine the numerators over the common denominator.

$f (\frac{1}{x}) = \sqrt{\frac{1 + 5 x}{x}}$

Step 5

Rewrite $\sqrt{\frac{1 + 5 x}{x}}$ as $\frac{\sqrt{1 + 5 x}}{\sqrt{x}}$ .

$f (\frac{1}{x}) = \frac{\sqrt{1 + 5 x}}{\sqrt{x}}$

Step 6

Multiply $\frac{\sqrt{1 + 5 x}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$f (\frac{1}{x}) = \frac{\sqrt{1 + 5 x}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{\sqrt{1 + 5 x}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$f (\frac{1}{x}) = \frac{\sqrt{1 + 5 x} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 7.2

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

$f (\frac{1}{x}) = \frac{\sqrt{1 + 5 x} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 7.3

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

$f (\frac{1}{x}) = \frac{\sqrt{1 + 5 x} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 7.4

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

$f (\frac{1}{x}) = \frac{\sqrt{1 + 5 x} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$f (\frac{1}{x}) = \frac{\sqrt{1 + 5 x} \sqrt{x}}{{\sqrt{x}}^{2}}$

Step 7.6

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

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Step 7.6.1

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

$f (\frac{1}{x}) = \frac{\sqrt{1 + 5 x} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}$

Step 7.6.2

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

$f (\frac{1}{x}) = \frac{\sqrt{1 + 5 x} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$f (\frac{1}{x}) = \frac{\sqrt{1 + 5 x} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$f (\frac{1}{x}) = \frac{\sqrt{1 + 5 x} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$f (\frac{1}{x}) = \frac{\sqrt{1 + 5 x} \sqrt{x}}{x}$

Step 7.6.5

Simplify.

$f (\frac{1}{x}) = \frac{\sqrt{1 + 5 x} \sqrt{x}}{x}$

Step 8

Combine using the product rule for radicals.

$f (\frac{1}{x}) = \frac{\sqrt{(1 + 5 x) x}}{x}$

Step 9

Reorder factors in $\frac{\sqrt{(1 + 5 x) x}}{x}$ .

$f (\frac{1}{x}) = \frac{\sqrt{x (1 + 5 x)}}{x}$