Algebra Examples

$f (x) = \frac{1 - x}{9 x}$ , $g (x) = \frac{1}{1 + 9 x}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{1}{1 + 9 x})$ by substituting in the value of $g$ into $f$ .

$f (\frac{1}{1 + 9 x}) = \frac{1 - (\frac{1}{1 + 9 x})}{9 (\frac{1}{1 + 9 x})}$

Step 3

Simplify the numerator.

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Step 3.1

Write $1$ as a fraction with a common denominator.

$f (\frac{1}{1 + 9 x}) = \frac{\frac{1 + 9 x}{1 + 9 x} - \frac{1}{1 + 9 x}}{9 (\frac{1}{1 + 9 x})}$

Step 3.2

Combine the numerators over the common denominator.

$f (\frac{1}{1 + 9 x}) = \frac{\frac{1 + 9 x - 1}{1 + 9 x}}{9 (\frac{1}{1 + 9 x})}$

Step 3.3

Reorder terms.

$f (\frac{1}{1 + 9 x}) = \frac{\frac{9 x + 1 - 1}{9 x + 1}}{9 (\frac{1}{1 + 9 x})}$

Step 3.4

Rewrite $\frac{9 x + 1 - 1}{9 x + 1}$ in a factored form.

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Step 3.4.1

Subtract $1$ from $1$ .

$f (\frac{1}{1 + 9 x}) = \frac{\frac{9 x + 0}{9 x + 1}}{9 (\frac{1}{1 + 9 x})}$

Step 3.4.2

Add $9 x$ and $0$ .

$f (\frac{1}{1 + 9 x}) = \frac{\frac{9 x}{9 x + 1}}{9 (\frac{1}{1 + 9 x})}$

Step 4

Combine $9$ and $\frac{1}{1 + 9 x}$ .

$f (\frac{1}{1 + 9 x}) = \frac{\frac{9 x}{9 x + 1}}{\frac{9}{1 + 9 x}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{1}{1 + 9 x}) = \frac{9 x}{9 x + 1} \cdot \frac{1 + 9 x}{9}$

Step 6

Cancel the common factor of $9$ .

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Step 6.1

Factor $9$ out of $9 x$ .

$f (\frac{1}{1 + 9 x}) = \frac{9 (x)}{9 x + 1} \cdot \frac{1 + 9 x}{9}$

Step 6.2

Cancel the common factor.

$f (\frac{1}{1 + 9 x}) = \frac{9 x}{9 x + 1} \cdot \frac{1 + 9 x}{9}$

Step 6.3

Rewrite the expression.

$f (\frac{1}{1 + 9 x}) = \frac{x}{9 x + 1} \cdot (1 + 9 x)$

Step 7

Multiply $\frac{x}{9 x + 1}$ by $1 + 9 x$ .

$f (\frac{1}{1 + 9 x}) = \frac{x (1 + 9 x)}{9 x + 1}$

Step 8

Cancel the common factor of $1 + 9 x$ and $9 x + 1$ .

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Step 8.1

Reorder terms.

$f (\frac{1}{1 + 9 x}) = \frac{x (9 x + 1)}{9 x + 1}$

Step 8.2

Cancel the common factor.

$f (\frac{1}{1 + 9 x}) = \frac{x (9 x + 1)}{9 x + 1}$

Step 8.3

Divide $x$ by $1$ .

$f (\frac{1}{1 + 9 x}) = x$