QuestionFind $g(f(x))$ given $f(x)=\frac{1}{2x+6}$ and $g(x)=\frac{4}{x}-3$ . Simplify fully, no decimals.

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is given by $f(x)=\frac{1}{ x+6}$ . The function $g(x)$ is given by $g(x)=\frac{4}{x}-3$
3. We are asked to find the composition of the functions $f$ and $g$ , denoted as $g(f(x))$

STEP 2

The composition of functions $f$ and $g$ is defined as $g(f(x))$ . This means we substitute $f(x)$ into $g(x)$ .
$g(f(x)) = g\left(\frac{1}{2x+6}\right)$

STEP 3

Now we substitute $f(x)$ into $g(x)$ .
$g(f(x)) = \frac{}{\frac{1}{2x+6}} -3$

STEP 4

implify the fraction within the fraction by multiplying the numerator and the denominator by the reciprocal of the denominator.
$g(f(x)) = \frac{4 \times (2x+6)}{1} -3$

STEP 5

implify the fraction.
$g(f(x)) =4 \times (2x+) -3$

STEP 6

istribute the4 across the terms in the parenthesis.
$g(f(x)) =8x +24 -3$

STEP 7

Combine like terms.
$g(f(x)) =x +21$ So, the composition of the functions $f$ and $g$ , denoted as $g(f(x))$ , is $x +21$ .

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