Algebra Examples

$f (x) = \frac{x - 5}{x + 8}$ $g (x) = \frac{x + 4}{x - 7}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{x + 4}{x - 7})$ by substituting in the value of $g$ into $f$ .

$f (\frac{x + 4}{x - 7}) = \frac{(\frac{x + 4}{x - 7}) - 5}{(\frac{x + 4}{x - 7}) + 8}$

Step 3

Multiply the numerator and denominator of the fraction by $x - 7$ .

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Step 3.1

Multiply $\frac{\frac{x + 4}{x - 7} - 5}{\frac{x + 4}{x - 7} + 8}$ by $\frac{x - 7}{x - 7}$ .

$f (\frac{x + 4}{x - 7}) = \frac{x - 7}{x - 7} \cdot \frac{\frac{x + 4}{x - 7} - 5}{\frac{x + 4}{x - 7} + 8}$

Step 3.2

Combine.

$f (\frac{x + 4}{x - 7}) = \frac{(x - 7) (\frac{x + 4}{x - 7} - 5)}{(x - 7) (\frac{x + 4}{x - 7} + 8)}$

Step 4

Apply the distributive property.

$f (\frac{x + 4}{x - 7}) = \frac{(x - 7) (\frac{x + 4}{x - 7}) + (x - 7) \cdot - 5}{(x - 7) (\frac{x + 4}{x - 7}) + (x - 7) \cdot 8}$

Step 5

Simplify by cancelling.

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Step 5.1

Cancel the common factor of $x - 7$ .

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Step 5.1.1

Cancel the common factor.

$f (\frac{x + 4}{x - 7}) = \frac{(x - 7) (\frac{x + 4}{x - 7}) + (x - 7) \cdot - 5}{(x - 7) (\frac{x + 4}{x - 7}) + (x - 7) \cdot 8}$

Step 5.1.2

Rewrite the expression.

$f (\frac{x + 4}{x - 7}) = \frac{x + 4 + (x - 7) \cdot - 5}{(x - 7) (\frac{x + 4}{x - 7}) + (x - 7) \cdot 8}$

Step 5.2

Cancel the common factor of $x - 7$ .

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Step 5.2.1

Cancel the common factor.

$f (\frac{x + 4}{x - 7}) = \frac{x + 4 + (x - 7) \cdot - 5}{(x - 7) (\frac{x + 4}{x - 7}) + (x - 7) \cdot 8}$

Step 5.2.2

Rewrite the expression.

$f (\frac{x + 4}{x - 7}) = \frac{x + 4 + (x - 7) \cdot - 5}{x + 4 + (x - 7) \cdot 8}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$f (\frac{x + 4}{x - 7}) = \frac{x + 4 + x \cdot - 5 - 7 \cdot - 5}{x + 4 + (x - 7) \cdot 8}$

Step 6.2

Move $- 5$ to the left of $x$ .

$f (\frac{x + 4}{x - 7}) = \frac{x + 4 - 5 \cdot x - 7 \cdot - 5}{x + 4 + (x - 7) \cdot 8}$

Step 6.3

Multiply $- 7$ by $- 5$ .

$f (\frac{x + 4}{x - 7}) = \frac{x + 4 - 5 \cdot x + 35}{x + 4 + (x - 7) \cdot 8}$

Step 6.4

Subtract $5 x$ from $x$ .

$f (\frac{x + 4}{x - 7}) = \frac{- 4 x + 4 + 35}{x + 4 + (x - 7) \cdot 8}$

Step 6.5

Add $4$ and $35$ .

$f (\frac{x + 4}{x - 7}) = \frac{- 4 x + 39}{x + 4 + (x - 7) \cdot 8}$

Step 7

Simplify the denominator.

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Step 7.1

Apply the distributive property.

$f (\frac{x + 4}{x - 7}) = \frac{- 4 x + 39}{x + 4 + x \cdot 8 - 7 \cdot 8}$

Step 7.2

Move $8$ to the left of $x$ .

$f (\frac{x + 4}{x - 7}) = \frac{- 4 x + 39}{x + 4 + 8 \cdot x - 7 \cdot 8}$

Step 7.3

Multiply $- 7$ by $8$ .

$f (\frac{x + 4}{x - 7}) = \frac{- 4 x + 39}{x + 4 + 8 \cdot x - 56}$

Step 7.4

Add $x$ and $8 x$ .

$f (\frac{x + 4}{x - 7}) = \frac{- 4 x + 39}{9 x + 4 - 56}$

Step 7.5

Subtract $56$ from $4$ .

$f (\frac{x + 4}{x - 7}) = \frac{- 4 x + 39}{9 x - 52}$

Step 8

Simplify with factoring out.

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Step 8.1

Factor $- 1$ out of $- 4 x$ .

$f (\frac{x + 4}{x - 7}) = \frac{- (4 x) + 39}{9 x - 52}$

Step 8.2

Rewrite $39$ as $- 1 (- 39)$ .

$f (\frac{x + 4}{x - 7}) = \frac{- (4 x) - 1 \cdot - 39}{9 x - 52}$

Step 8.3

Factor $- 1$ out of $- (4 x) - 1 (- 39)$ .

$f (\frac{x + 4}{x - 7}) = \frac{- (4 x - 39)}{9 x - 52}$

Step 8.4

Simplify the expression.

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Step 8.4.1

Rewrite $- (4 x - 39)$ as $- 1 (4 x - 39)$ .

$f (\frac{x + 4}{x - 7}) = \frac{- 1 (4 x - 39)}{9 x - 52}$

Step 8.4.2

Move the negative in front of the fraction.

$f (\frac{x + 4}{x - 7}) = - \frac{4 x - 39}{9 x - 52}$