Finite Math Examples

$f (x) = \frac{x}{x - 7}$ , $g (x) = \frac{- 3}{x}$ , $(f \circ g)$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

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

$f (\frac{- 3}{x}) = \frac{\frac{- 3}{x}}{(\frac{- 3}{x}) - 7}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{- 3}{x}) = \frac{- 3}{x} \cdot \frac{1}{\frac{- 3}{x} - 7}$

Step 4

Move the negative in front of the fraction.

$f (\frac{- 3}{x}) = - \frac{3}{x} \cdot \frac{1}{\frac{- 3}{x} - 7}$

Step 5

Simplify the denominator.

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Step 5.1

Move the negative in front of the fraction.

$f (\frac{- 3}{x}) = - \frac{3}{x} \cdot \frac{1}{- \frac{3}{x} - 7}$

Step 5.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$f (\frac{- 3}{x}) = - \frac{3}{x} \cdot \frac{1}{- \frac{3}{x} - 7 \cdot \frac{x}{x}}$

Step 5.3

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

$f (\frac{- 3}{x}) = - \frac{3}{x} \cdot \frac{1}{- \frac{3}{x} + \frac{- 7 x}{x}}$

Step 5.4

Combine the numerators over the common denominator.

$f (\frac{- 3}{x}) = - \frac{3}{x} \cdot \frac{1}{\frac{- 3 - 7 x}{x}}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{- 3}{x}) = - \frac{3}{x} \cdot (1 (\frac{x}{- 3 - 7 x}))$

Step 7

Multiply $\frac{x}{- 3 - 7 x}$ by $1$ .

$f (\frac{- 3}{x}) = - \frac{3}{x} \cdot \frac{x}{- 3 - 7 x}$

Step 8

Cancel the common factor of $x$ .

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Step 8.1

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

$f (\frac{- 3}{x}) = \frac{- 3}{x} \cdot \frac{x}{- 3 - 7 x}$

Step 8.2

Cancel the common factor.

$f (\frac{- 3}{x}) = \frac{- 3}{x} \cdot \frac{x}{- 3 - 7 x}$

Step 8.3

Rewrite the expression.

$f (\frac{- 3}{x}) = - 3 \frac{1}{- 3 - 7 x}$

Step 9

Combine $- 3$ and $\frac{1}{- 3 - 7 x}$ .

$f (\frac{- 3}{x}) = \frac{- 3}{- 3 - 7 x}$

Step 10

Move the negative in front of the fraction.

$f (\frac{- 3}{x}) = - \frac{3}{- 3 - 7 x}$

Step 11

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

$f (\frac{- 3}{x}) = - \frac{3}{- 1 \cdot 3 - 7 x}$

Step 12

Factor $- 1$ out of $- 7 x$ .

$f (\frac{- 3}{x}) = - \frac{3}{- 1 \cdot 3 - (7 x)}$

Step 13

Factor $- 1$ out of $- 1 (3) - (7 x)$ .

$f (\frac{- 3}{x}) = - \frac{3}{- 1 (3 + 7 x)}$

Step 14

Simplify the expression.

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Step 14.1

Move the negative in front of the fraction.

$f (\frac{- 3}{x}) = \frac{3}{3 + 7 x}$

Step 14.2

Multiply $- 1$ by $- 1$ .

$f (\frac{- 3}{x}) = 1 (\frac{3}{3 + 7 x})$

Step 14.3

Multiply $\frac{3}{3 + 7 x}$ by $1$ .

$f (\frac{- 3}{x}) = \frac{3}{3 + 7 x}$