Precalculus Examples

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

Step 1

Set up the composite result function.

$f (f (x))$

Step 2

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

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

Step 3

Remove parentheses.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 5.3

Add $x$ and $x$ .

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

Step 6

Multiply the numerator by the reciprocal of the denominator.

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

Step 7

Multiply $\frac{1 + x}{1 + 2 x}$ by $1$ .

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

Step 8

Cancel the common factor of $1 + x$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

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

Step 9

Combine $x$ and $\frac{1}{1 + 2 x}$ .

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