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

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

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

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

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

Step 3

Simplify the denominator.

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

$g (\frac{2 - x}{x}) = \frac{2}{\frac{2 - x}{x} + \frac{x}{x}}$

Combine the numerators over the common denominator.

$g (\frac{2 - x}{x}) = \frac{2}{\frac{2 - x + x}{x}}$

Rewrite $\frac{2 - x + x}{x}$ in a factored form.

Add $- x$ and $x$ .

$g (\frac{2 - x}{x}) = \frac{2}{\frac{2 + 0}{x}}$

Add $2$ and $0$ .

$g (\frac{2 - x}{x}) = \frac{2}{\frac{2}{x}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$g (\frac{2 - x}{x}) = 2 (\frac{x}{2})$

Step 5

Cancel the common factor of $2$ .

Cancel the common factor.

$g (\frac{2 - x}{x}) = 2 (\frac{x}{2})$

Rewrite the expression.

$g (\frac{2 - x}{x}) = x$