Algebra Examples

$f (x) = \frac{x - 1}{x}$ $g (x) = \frac{x}{x + 2}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

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

$g (\frac{x - 1}{x}) = \frac{\frac{x - 1}{x}}{(\frac{x - 1}{x}) + 2}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$g (\frac{x - 1}{x}) = \frac{x - 1}{x} \cdot \frac{1}{\frac{x - 1}{x} + 2}$

Step 4

Simplify the denominator.

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Step 4.1

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

$g (\frac{x - 1}{x}) = \frac{x - 1}{x} \cdot \frac{1}{\frac{x - 1}{x} + \frac{2 x}{x}}$

Step 4.2

Combine the numerators over the common denominator.

$g (\frac{x - 1}{x}) = \frac{x - 1}{x} \cdot \frac{1}{\frac{x - 1 + 2 x}{x}}$

Step 4.3

Reorder terms.

$g (\frac{x - 1}{x}) = \frac{x - 1}{x} \cdot \frac{1}{\frac{2 x + x - 1}{x}}$

Step 4.4

Add $2 x$ and $x$ .

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

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

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

Step 7

Cancel the common factor of $x$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

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

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