Algebra Examples

$f (x) = \sqrt{x - 5}$ , $g (x) = \frac{x + 2}{3}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

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

$f (\frac{x + 2}{3}) = \sqrt{(\frac{x + 2}{3}) - 5}$

Step 3

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{x + 2}{3}) = \sqrt{\frac{x + 2}{3} - 5 \cdot \frac{3}{3}}$

Step 4

Combine $- 5$ and $\frac{3}{3}$ .

$f (\frac{x + 2}{3}) = \sqrt{\frac{x + 2}{3} + \frac{- 5 \cdot 3}{3}}$

Step 5

Combine the numerators over the common denominator.

$f (\frac{x + 2}{3}) = \sqrt{\frac{x + 2 - 5 \cdot 3}{3}}$

Step 6

Rewrite $\frac{x + 2 - 5 \cdot 3}{3}$ in a factored form.

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Step 6.1

Multiply $- 5$ by $3$ .

$f (\frac{x + 2}{3}) = \sqrt{\frac{x + 2 - 15}{3}}$

Step 6.2

Subtract $15$ from $2$ .

$f (\frac{x + 2}{3}) = \sqrt{\frac{x - 13}{3}}$

Step 7

Rewrite $\sqrt{\frac{x - 13}{3}}$ as $\frac{\sqrt{x - 13}}{\sqrt{3}}$ .

$f (\frac{x + 2}{3}) = \frac{\sqrt{x - 13}}{\sqrt{3}}$

Step 8

Multiply $\frac{\sqrt{x - 13}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (\frac{x + 2}{3}) = \frac{\sqrt{x - 13}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{\sqrt{x - 13}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (\frac{x + 2}{3}) = \frac{\sqrt{x - 13} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 9.2

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{x + 2}{3}) = \frac{\sqrt{x - 13} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 9.3

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{x + 2}{3}) = \frac{\sqrt{x - 13} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 9.4

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

$f (\frac{x + 2}{3}) = \frac{\sqrt{x - 13} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 9.5

Add $1$ and $1$ .

$f (\frac{x + 2}{3}) = \frac{\sqrt{x - 13} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 9.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 9.6.1

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

$f (\frac{x + 2}{3}) = \frac{\sqrt{x - 13} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 9.6.2

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

$f (\frac{x + 2}{3}) = \frac{\sqrt{x - 13} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 9.6.3

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

$f (\frac{x + 2}{3}) = \frac{\sqrt{x - 13} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

$f (\frac{x + 2}{3}) = \frac{\sqrt{x - 13} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 9.6.4.2

Rewrite the expression.

$f (\frac{x + 2}{3}) = \frac{\sqrt{x - 13} \sqrt{3}}{3}$

Step 9.6.5

Evaluate the exponent.

$f (\frac{x + 2}{3}) = \frac{\sqrt{x - 13} \sqrt{3}}{3}$

Step 10

Combine using the product rule for radicals.

$f (\frac{x + 2}{3}) = \frac{\sqrt{(x - 13) \cdot 3}}{3}$

Step 11

Reorder factors in $\frac{\sqrt{(x - 13) \cdot 3}}{3}$ .

$f (\frac{x + 2}{3}) = \frac{\sqrt{3 (x - 13)}}{3}$