Algebra Examples

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

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

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

Step 3

Remove parentheses.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Simplify the denominator.

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Step 5.1

To write $9$ as a fraction with a common denominator, multiply by $\frac{1 - x}{1 - x}$ .

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 5.3

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

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Step 5.3.1

Factor $9$ out of $9 (1 - x) + 9 x$ .

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Step 5.3.1.1

Factor $9$ out of $9 x$ .

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

Step 5.3.1.2

Factor $9$ out of $9 (1 - x) + 9 (x)$ .

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

Step 5.3.2

Add $- x$ and $x$ .

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

Step 5.3.3

Add $1$ and $0$ .

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

Step 5.4

Multiply $9$ by $1$ .

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

Step 6

Multiply the numerator by the reciprocal of the denominator.

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

Step 7

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

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

Step 8

Cancel the common factor of $9$ .

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Step 8.1

Factor $9$ out of $9 x$ .

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

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

Step 9

Cancel the common factor of $1 - x$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

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