Algebra Examples

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (- \frac{5}{x})$ by substituting in the value of $g$ into $f$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{5}{x}) = - \frac{5}{x} \cdot \frac{1}{- \frac{5}{x} - 3}$

Step 4

Simplify the denominator.

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Step 4.1

Factor $- 1$ out of $- \frac{5}{x} - 3$ .

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Step 4.1.1

Factor $- 1$ out of $- \frac{5}{x}$ .

$f (- \frac{5}{x}) = - \frac{5}{x} \cdot \frac{1}{- (\frac{5}{x}) - 3}$

Step 4.1.2

Rewrite $- 3$ as $- 1 (3)$ .

$f (- \frac{5}{x}) = - \frac{5}{x} \cdot \frac{1}{- (\frac{5}{x}) - 1 \cdot 3}$

Step 4.1.3

Factor $- 1$ out of $- (\frac{5}{x}) - 1 (3)$ .

$f (- \frac{5}{x}) = - \frac{5}{x} \cdot \frac{1}{- (\frac{5}{x} + 3)}$

Step 4.2

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

$f (- \frac{5}{x}) = - \frac{5}{x} \cdot \frac{1}{- (\frac{5}{x} + \frac{3 x}{x})}$

Step 4.3

Combine the numerators over the common denominator.

$f (- \frac{5}{x}) = - \frac{5}{x} \cdot \frac{1}{- \frac{5 + 3 x}{x}}$

Step 5

Cancel the common factor of $1$ and $- 1$ .

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Step 5.1

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

$f (- \frac{5}{x}) = - \frac{5}{x} \cdot \frac{- 1 \cdot - 1}{- \frac{5 + 3 x}{x}}$

Step 5.2

Move the negative in front of the fraction.

$f (- \frac{5}{x}) = - \frac{5}{x} \cdot (- \frac{1}{\frac{5 + 3 x}{x}})$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{5}{x}) = - \frac{5}{x} \cdot (- (1 (\frac{x}{5 + 3 x})))$

Step 7

Multiply $\frac{x}{5 + 3 x}$ by $1$ .

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

Step 8

Cancel the common factor of $x$ .

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Step 8.1

Move the leading negative in $- \frac{5}{x}$ into the numerator.

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

Step 8.2

Move the leading negative in $- \frac{x}{5 + 3 x}$ into the numerator.

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

Step 8.3

Factor $x$ out of $- x$ .

$f (- \frac{5}{x}) = \frac{- 5}{x} \cdot \frac{x \cdot - 1}{5 + 3 x}$

Step 8.4

Cancel the common factor.

$f (- \frac{5}{x}) = \frac{- 5}{x} \cdot \frac{x \cdot - 1}{5 + 3 x}$

Step 8.5

Rewrite the expression.

$f (- \frac{5}{x}) = - 5 \frac{- 1}{5 + 3 x}$

Step 9

Combine $- 5$ and $\frac{- 1}{5 + 3 x}$ .

$f (- \frac{5}{x}) = \frac{- 5 \cdot - 1}{5 + 3 x}$

Step 10

Multiply $- 5$ by $- 1$ .

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