Questions: Find (f circ g)(x) for the indicated functions and simplify. f(x)=8 x+4, g(x)=6/(x+4) (f circ g)(x)= (Simplify your answer.)

Transcript text: Find $(f \circ g)(x)$ for the indicated functions and simplify. \[ \begin{array}{l} f(x)=8 x+4, g(x)=\frac{6}{x+4} \\ (f \circ g)(x)=\square \end{array} \] (Simplify your answer.)

Solution

Solution Steps

To find $(f \circ g)(x)$, we need to substitute $g(x)$ into $f(x)$. This means replacing every instance of $x$ in $f(x)$ with $g(x)$. After substitution, simplify the resulting expression.

Step 1: Substitute $ g(x) $ into $ f(x) $

To find $(f \circ g)(x)$, we substitute $g(x) = \frac{6}{x+4}$ into $f(x) = 8x + 4$. This gives us:

\[ f(g(x)) = 8\left(\frac{6}{x+4}\right) + 4 \]

Step 2: Simplify the Expression

Simplify the expression obtained from the substitution:

\[ f(g(x)) = \frac{48}{x+4} + 4 \]

Combine the terms over a common denominator:

\[ f(g(x)) = \frac{48 + 4(x+4)}{x+4} = \frac{48 + 4x + 16}{x+4} = \frac{4x + 64}{x+4} \]

Factor out the common factor in the numerator:

\[ f(g(x)) = \frac{4(x + 16)}{x+4} \]