Algebra Examples

$f (x) = \frac{2 x + 4}{x + 9}$ $g (x) = \frac{9 x - 4}{2 - x}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{9 x - 4}{2 - x})$ by substituting in the value of $g$ into $f$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{2 (\frac{9 x - 4}{2 - x}) + 4}{(\frac{9 x - 4}{2 - x}) + 9}$

Step 3

Multiply the numerator and denominator of the fraction by $2 - x$ .

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Step 3.1

Multiply $\frac{2 \frac{9 x - 4}{2 - x} + 4}{\frac{9 x - 4}{2 - x} + 9}$ by $\frac{2 - x}{2 - x}$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{2 - x}{2 - x} \cdot \frac{2 (\frac{9 x - 4}{2 - x}) + 4}{\frac{9 x - 4}{2 - x} + 9}$

Step 3.2

Combine.

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) (2 (\frac{9 x - 4}{2 - x}) + 4)}{(2 - x) (\frac{9 x - 4}{2 - x} + 9)}$

Step 4

Apply the distributive property.

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) (2 (\frac{9 x - 4}{2 - x})) + (2 - x) \cdot 4}{(2 - x) (\frac{9 x - 4}{2 - x}) + (2 - x) \cdot 9}$

Step 5

Cancel the common factor of $2 - x$ .

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Step 5.1

Cancel the common factor.

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) (2 (\frac{9 x - 4}{2 - x})) + (2 - x) \cdot 4}{(2 - x) (\frac{9 x - 4}{2 - x}) + (2 - x) \cdot 9}$

Step 5.2

Rewrite the expression.

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) (2 (\frac{9 x - 4}{2 - x})) + (2 - x) \cdot 4}{9 x - 4 + (2 - x) \cdot 9}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $(2 - x) \cdot 2$ out of $(2 - x) \cdot 2 \frac{9 x - 4}{2 - x} + (2 - x) \cdot 4$ .

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Step 6.1.1

Factor $(2 - x) \cdot 2$ out of $(2 - x) \cdot 2 \frac{9 x - 4}{2 - x}$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot (2 (\frac{9 x - 4}{2 - x})) + (2 - x) \cdot 4}{9 x - 4 + (2 - x) \cdot 9}$

Step 6.1.2

Factor $(2 - x) \cdot 2$ out of $(2 - x) \cdot 4$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot (2 (\frac{9 x - 4}{2 - x})) + (2 - x) \cdot 2 \cdot 2}{9 x - 4 + (2 - x) \cdot 9}$

Step 6.1.3

Factor $(2 - x) \cdot 2$ out of $(2 - x) \cdot 2 \frac{9 x - 4}{2 - x} + (2 - x) \cdot 2 \cdot 2$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot (2 (\frac{9 x - 4}{2 - x} + 2))}{9 x - 4 + (2 - x) \cdot 9}$

Step 6.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{2 - x}{2 - x}$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot (2 (\frac{9 x - 4}{2 - x} + \frac{2 (2 - x)}{2 - x}))}{9 x - 4 + (2 - x) \cdot 9}$

Step 6.3

Combine the numerators over the common denominator.

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot (2 (\frac{9 x - 4 + 2 (2 - x)}{2 - x}))}{9 x - 4 + (2 - x) \cdot 9}$

Step 6.4

Reorder terms.

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot (2 (\frac{9 x + 2 (2 - x) - 4}{- x + 2}))}{9 x - 4 + (2 - x) \cdot 9}$

Step 6.5

Rewrite $\frac{9 x + 2 (2 - x) - 4}{- x + 2}$ in a factored form.

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Step 6.5.1

Apply the distributive property.

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot (2 (\frac{9 x + 2 \cdot 2 + 2 (- x) - 4}{- x + 2}))}{9 x - 4 + (2 - x) \cdot 9}$

Step 6.5.2

Multiply $2$ by $2$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot (2 (\frac{9 x + 4 + 2 (- x) - 4}{- x + 2}))}{9 x - 4 + (2 - x) \cdot 9}$

Step 6.5.3

Multiply $- 1$ by $2$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot (2 (\frac{9 x + 4 - 2 x - 4}{- x + 2}))}{9 x - 4 + (2 - x) \cdot 9}$

Step 6.5.4

Subtract $2 x$ from $9 x$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot (2 (\frac{7 x + 4 - 4}{- x + 2}))}{9 x - 4 + (2 - x) \cdot 9}$

Step 6.5.5

Subtract $4$ from $4$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot (2 (\frac{7 x + 0}{- x + 2}))}{9 x - 4 + (2 - x) \cdot 9}$

Step 6.5.6

Add $7 x$ and $0$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot (2 (\frac{7 x}{- x + 2}))}{9 x - 4 + (2 - x) \cdot 9}$

Step 6.6

Combine exponents.

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Step 6.6.1

Combine $\frac{7 x}{- x + 2}$ and $2$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot \frac{7 x \cdot 2}{- x + 2}}{9 x - 4 + (2 - x) \cdot 9}$

Step 6.6.2

Multiply $2$ by $7$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot \frac{14 x}{- x + 2}}{9 x - 4 + (2 - x) \cdot 9}$

Step 7

Simplify the denominator.

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Step 7.1

Apply the distributive property.

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot \frac{14 x}{- x + 2}}{9 x - 4 + 2 \cdot 9 - x \cdot 9}$

Step 7.2

Multiply $2$ by $9$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot \frac{14 x}{- x + 2}}{9 x - 4 + 18 - x \cdot 9}$

Step 7.3

Multiply $9$ by $- 1$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot \frac{14 x}{- x + 2}}{9 x - 4 + 18 - 9 x}$

Step 7.4

Subtract $9 x$ from $9 x$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot \frac{14 x}{- x + 2}}{0 - 4 + 18}$

Step 7.5

Subtract $4$ from $0$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot \frac{14 x}{- x + 2}}{- 4 + 18}$

Step 7.6

Add $- 4$ and $18$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{(2 - x) \cdot \frac{14 x}{- x + 2}}{14}$

Step 8

Simplify terms.

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Step 8.1

Factor $\frac{14 x}{- x + 2}$ out of $\frac{(2 - x) \cdot \frac{14 x}{- x + 2}}{14}$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{14 x}{- x + 2} \cdot \frac{2 - x}{14}$

Step 8.2

Cancel the common factor of $14$ .

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Step 8.2.1

Factor $14$ out of $14 x$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{14 (x)}{- x + 2} \cdot \frac{2 - x}{14}$

Step 8.2.2

Cancel the common factor.

$f (\frac{9 x - 4}{2 - x}) = \frac{14 x}{- x + 2} \cdot \frac{2 - x}{14}$

Step 8.2.3

Rewrite the expression.

$f (\frac{9 x - 4}{2 - x}) = \frac{x}{- x + 2} \cdot (2 - x)$

Step 8.3

Multiply $\frac{x}{- x + 2}$ by $2 - x$ .

$f (\frac{9 x - 4}{2 - x}) = \frac{x (2 - x)}{- x + 2}$

Step 8.4

Cancel the common factor of $2 - x$ and $- x + 2$ .

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Step 8.4.1

Reorder terms.

$f (\frac{9 x - 4}{2 - x}) = \frac{x (- x + 2)}{- x + 2}$

Step 8.4.2

Cancel the common factor.

$f (\frac{9 x - 4}{2 - x}) = \frac{x (- x + 2)}{- x + 2}$

Step 8.4.3

Divide $x$ by $1$ .

$f (\frac{9 x - 4}{2 - x}) = x$