Find f(g(x)) f(x)=x/(2+x) g(x)=(5x)/(1-x)

$f (x) = \frac{x}{2 + x}$ $g (x) = \frac{5 x}{1 - x}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{5 x}{1 - x})$ by substituting in the value of $g$ into $f$ .

$f (\frac{5 x}{1 - x}) = \frac{\frac{5 x}{1 - x}}{2 + \frac{5 x}{1 - x}}$

Step 3

Remove parentheses.

$f (\frac{5 x}{1 - x}) = \frac{\frac{5 x}{1 - x}}{2 + \frac{5 x}{1 - x}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{5 x}{1 - x}) = \frac{5 x}{1 - x} \cdot \frac{1}{2 + \frac{5 x}{1 - x}}$

Step 5

Simplify the denominator.

To write $2$ as a fraction with a common denominator, multiply by $\frac{1 - x}{1 - x}$ .

$f (\frac{5 x}{1 - x}) = \frac{5 x}{1 - x} \cdot \frac{1}{\frac{2 (1 - x)}{1 - x} + \frac{5 x}{1 - x}}$

Combine the numerators over the common denominator.

$f (\frac{5 x}{1 - x}) = \frac{5 x}{1 - x} \cdot \frac{1}{\frac{2 (1 - x) + 5 x}{1 - x}}$

Rewrite $\frac{2 (1 - x) + 5 x}{1 - x}$ in a factored form.

Apply the distributive property.

$f (\frac{5 x}{1 - x}) = \frac{5 x}{1 - x} \cdot \frac{1}{\frac{2 \cdot 1 + 2 (- x) + 5 x}{1 - x}}$

Multiply $2$ by $1$ .

$f (\frac{5 x}{1 - x}) = \frac{5 x}{1 - x} \cdot \frac{1}{\frac{2 + 2 (- x) + 5 x}{1 - x}}$

Multiply $- 1$ by $2$ .

$f (\frac{5 x}{1 - x}) = \frac{5 x}{1 - x} \cdot \frac{1}{\frac{2 - 2 x + 5 x}{1 - x}}$

Add $- 2 x$ and $5 x$ .

$f (\frac{5 x}{1 - x}) = \frac{5 x}{1 - x} \cdot \frac{1}{\frac{2 + 3 x}{1 - x}}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{5 x}{1 - x}) = \frac{5 x}{1 - x} \cdot (1 (\frac{1 - x}{2 + 3 x}))$

Step 7

Multiply $\frac{1 - x}{2 + 3 x}$ by $1$ .

$f (\frac{5 x}{1 - x}) = \frac{5 x}{1 - x} \cdot \frac{1 - x}{2 + 3 x}$

Step 8

Cancel the common factor of $1 - x$ .

Cancel the common factor.

$f (\frac{5 x}{1 - x}) = \frac{5 x}{1 - x} \cdot \frac{1 - x}{2 + 3 x}$

Rewrite the expression.

$f (\frac{5 x}{1 - x}) = 5 x (\frac{1}{2 + 3 x})$

Step 9

Combine $5$ and $\frac{1}{2 + 3 x}$ .

$f (\frac{5 x}{1 - x}) = x (\frac{5}{2 + 3 x})$

Step 10

Combine $x$ and $\frac{5}{2 + 3 x}$ .

$f (\frac{5 x}{1 - x}) = \frac{x \cdot 5}{2 + 3 x}$

Step 11

Move $5$ to the left of $x$ .

$f (\frac{5 x}{1 - x}) = \frac{5 x}{2 + 3 x}$

Find f(g(x)) f(x)=x/(2+x) g(x)=(5x)/(1-x)

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