Find f(g(x)) f(x)=7x-5 , g(x)=6/(x+5)

$f (x) = 7 x - 5$ , $g (x) = \frac{6}{x + 5}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{6}{x + 5})$ by substituting in the value of $g$ into $f$ .

$f (\frac{6}{x + 5}) = 7 (\frac{6}{x + 5}) - 5$

Step 3

Multiply $7 (\frac{6}{x + 5})$ .

Combine $7$ and $\frac{6}{x + 5}$ .

$f (\frac{6}{x + 5}) = \frac{7 \cdot 6}{x + 5} - 5$

Multiply $7$ by $6$ .

$f (\frac{6}{x + 5}) = \frac{42}{x + 5} - 5$

Step 4

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$f (\frac{6}{x + 5}) = \frac{42}{x + 5} - 5 \cdot \frac{x + 5}{x + 5}$

Step 5

Simplify terms.

Combine $- 5$ and $\frac{x + 5}{x + 5}$ .

$f (\frac{6}{x + 5}) = \frac{42}{x + 5} + \frac{- 5 (x + 5)}{x + 5}$

Combine the numerators over the common denominator.

$f (\frac{6}{x + 5}) = \frac{42 - 5 (x + 5)}{x + 5}$

Step 6

Simplify the numerator.

Apply the distributive property.

$f (\frac{6}{x + 5}) = \frac{42 - 5 x - 5 \cdot 5}{x + 5}$

Multiply $- 5$ by $5$ .

$f (\frac{6}{x + 5}) = \frac{42 - 5 x - 25}{x + 5}$

Subtract $25$ from $42$ .

$f (\frac{6}{x + 5}) = \frac{- 5 x + 17}{x + 5}$

Step 7

Simplify with factoring out.

Factor $- 1$ out of $- 5 x$ .

$f (\frac{6}{x + 5}) = \frac{- (5 x) + 17}{x + 5}$

Rewrite $17$ as $- 1 (- 17)$ .

$f (\frac{6}{x + 5}) = \frac{- (5 x) - 1 \cdot - 17}{x + 5}$

Factor $- 1$ out of $- (5 x) - 1 (- 17)$ .

$f (\frac{6}{x + 5}) = \frac{- (5 x - 17)}{x + 5}$

Simplify the expression.

Rewrite $- (5 x - 17)$ as $- 1 (5 x - 17)$ .

$f (\frac{6}{x + 5}) = \frac{- 1 (5 x - 17)}{x + 5}$

Move the negative in front of the fraction.

$f (\frac{6}{x + 5}) = - \frac{5 x - 17}{x + 5}$