Find g(f(x)) f(x)=x/(4+7x) , g(x)=x/(4-7x)

$f (x) = \frac{x}{4 + 7 x}$ , $g (x) = \frac{x}{4 - 7 x}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\frac{x}{4 + 7 x})$ by substituting in the value of $f$ into $g$ .

$g (\frac{x}{4 + 7 x}) = \frac{\frac{x}{4 + 7 x}}{4 - 7 \frac{x}{4 + 7 x}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$g (\frac{x}{4 + 7 x}) = \frac{x}{4 + 7 x} \cdot \frac{1}{4 - 7 \frac{x}{4 + 7 x}}$

Step 4

Simplify the denominator.

Combine $- 7$ and $\frac{x}{4 + 7 x}$ .

$g (\frac{x}{4 + 7 x}) = \frac{x}{4 + 7 x} \cdot \frac{1}{4 + \frac{- 7 x}{4 + 7 x}}$

Move the negative in front of the fraction.

$g (\frac{x}{4 + 7 x}) = \frac{x}{4 + 7 x} \cdot \frac{1}{4 - \frac{(7) x}{4 + 7 x}}$

To write $4$ as a fraction with a common denominator, multiply by $\frac{4 + 7 x}{4 + 7 x}$ .

$g (\frac{x}{4 + 7 x}) = \frac{x}{4 + 7 x} \cdot \frac{1}{\frac{4 (4 + 7 x)}{4 + 7 x} - \frac{7 x}{4 + 7 x}}$

Combine the numerators over the common denominator.

$g (\frac{x}{4 + 7 x}) = \frac{x}{4 + 7 x} \cdot \frac{1}{\frac{4 (4 + 7 x) - 7 x}{4 + 7 x}}$

Rewrite $\frac{4 (4 + 7 x) - 7 x}{4 + 7 x}$ in a factored form.

Apply the distributive property.

$g (\frac{x}{4 + 7 x}) = \frac{x}{4 + 7 x} \cdot \frac{1}{\frac{4 \cdot 4 + 4 (7 x) - 7 x}{4 + 7 x}}$

Multiply $4$ by $4$ .

$g (\frac{x}{4 + 7 x}) = \frac{x}{4 + 7 x} \cdot \frac{1}{\frac{16 + 4 (7 x) - 7 x}{4 + 7 x}}$

Multiply $7$ by $4$ .

$g (\frac{x}{4 + 7 x}) = \frac{x}{4 + 7 x} \cdot \frac{1}{\frac{16 + 28 x - 7 x}{4 + 7 x}}$

Subtract $7 x$ from $28 x$ .

$g (\frac{x}{4 + 7 x}) = \frac{x}{4 + 7 x} \cdot \frac{1}{\frac{16 + 21 x}{4 + 7 x}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$g (\frac{x}{4 + 7 x}) = \frac{x}{4 + 7 x} \cdot (1 (\frac{4 + 7 x}{16 + 21 x}))$

Step 6

Multiply $\frac{4 + 7 x}{16 + 21 x}$ by $1$ .

$g (\frac{x}{4 + 7 x}) = \frac{x}{4 + 7 x} \cdot \frac{4 + 7 x}{16 + 21 x}$

Step 7

Cancel the common factor of $4 + 7 x$ .

Cancel the common factor.

$g (\frac{x}{4 + 7 x}) = \frac{x}{4 + 7 x} \cdot \frac{4 + 7 x}{16 + 21 x}$

Rewrite the expression.

$g (\frac{x}{4 + 7 x}) = x (\frac{1}{16 + 21 x})$

Step 8

Combine $x$ and $\frac{1}{16 + 21 x}$ .

$g (\frac{x}{4 + 7 x}) = \frac{x}{16 + 21 x}$