Find f(g(x)) f(x)=(1-x)/x , g(x)=1/(1+x^2)

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

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.

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

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

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.

Subtract $1$ from $1$ .

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

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 $1 + x^{2}$ .

Cancel the common factor.

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

Rewrite the expression.

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