Algebra Examples

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{1}{1 + x^{2}})$ by substituting in the value of $g$ into $f$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Combine into one fraction.

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Step 4.1

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

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

Step 4.2

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Subtract $1$ from $1$ .

$f (\frac{1}{1 + x^{2}}) = \frac{x^{2} + 0}{1 + x^{2}} \cdot (1 + x^{2})$

Step 5.2

Add $x^{2}$ and $0$ .

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

Step 6

Cancel the common factor of $1 + x^{2}$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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