Algebra Examples

$f (x) = \frac{x}{x + 9}$ ; $g (x) = \frac{27}{x + 1}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{27}{x + 1})$ by substituting in the value of $g$ into $f$ .

$f (\frac{27}{x + 1}) = \frac{\frac{27}{x + 1}}{(\frac{27}{x + 1}) + 9}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{27}{x + 1}) = \frac{27}{x + 1} \cdot \frac{1}{\frac{27}{x + 1} + 9}$

Step 4

Simplify the denominator.

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Step 4.1

To write $9$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$f (\frac{27}{x + 1}) = \frac{27}{x + 1} \cdot \frac{1}{\frac{27}{x + 1} + \frac{9 (x + 1)}{x + 1}}$

Step 4.2

Combine the numerators over the common denominator.

$f (\frac{27}{x + 1}) = \frac{27}{x + 1} \cdot \frac{1}{\frac{27 + 9 (x + 1)}{x + 1}}$

Step 4.3

Reorder terms.

$f (\frac{27}{x + 1}) = \frac{27}{x + 1} \cdot \frac{1}{\frac{9 (x + 1) + 27}{x + 1}}$

Step 4.4

Rewrite $\frac{9 (x + 1) + 27}{x + 1}$ in a factored form.

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Step 4.4.1

Factor $9$ out of $9 (x + 1) + 27$ .

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Step 4.4.1.1

Factor $9$ out of $27$ .

$f (\frac{27}{x + 1}) = \frac{27}{x + 1} \cdot \frac{1}{\frac{9 (x + 1) + 9 \cdot 3}{x + 1}}$

Step 4.4.1.2

Factor $9$ out of $9 (x + 1) + 9 \cdot 3$ .

$f (\frac{27}{x + 1}) = \frac{27}{x + 1} \cdot \frac{1}{\frac{9 (x + 1 + 3)}{x + 1}}$

Step 4.4.2

Add $1$ and $3$ .

$f (\frac{27}{x + 1}) = \frac{27}{x + 1} \cdot \frac{1}{\frac{9 (x + 4)}{x + 1}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{27}{x + 1}) = \frac{27}{x + 1} \cdot (1 (\frac{x + 1}{9 (x + 4)}))$

Step 6

Multiply $\frac{x + 1}{9 (x + 4)}$ by $1$ .

$f (\frac{27}{x + 1}) = \frac{27}{x + 1} \cdot \frac{x + 1}{9 (x + 4)}$

Step 7

Cancel the common factor of $9$ .

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Step 7.1

Factor $9$ out of $27$ .

$f (\frac{27}{x + 1}) = \frac{9 (3)}{x + 1} \cdot \frac{x + 1}{9 (x + 4)}$

Step 7.2

Cancel the common factor.

$f (\frac{27}{x + 1}) = \frac{9 \cdot 3}{x + 1} \cdot \frac{x + 1}{9 (x + 4)}$

Step 7.3

Rewrite the expression.

$f (\frac{27}{x + 1}) = \frac{3}{x + 1} \cdot \frac{x + 1}{x + 4}$

Step 8

Cancel the common factor of $x + 1$ .

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Step 8.1

Cancel the common factor.

$f (\frac{27}{x + 1}) = \frac{3}{x + 1} \cdot \frac{x + 1}{x + 4}$

Step 8.2

Rewrite the expression.

$f (\frac{27}{x + 1}) = 3 (\frac{1}{x + 4})$

Step 9

Combine $3$ and $\frac{1}{x + 4}$ .

$f (\frac{27}{x + 1}) = \frac{3}{x + 4}$