Questions: For f(x)=4/(x+8) and g(x)=5/x, find a. (f ∘ g)(x); b. the domain of f ∘ g a. (f ∘ g)(x)=

Transcript text: For $f(x)=\frac{4}{x+8}$ and $g(x)=\frac{5}{x}$, find a. $(\mathrm{f} \circ \mathrm{g})(\mathrm{x})$; b. the domain of $f \circ g$ a. $(f \circ g)(x)=$ $\square$ (Simplify your answer.)

Solution

Solution Steps

Solution Approach

To find $(f \circ g)(x)$, substitute $g(x)$ into $f(x)$. This means replacing $x$ in $f(x)$ with $g(x)$.
Simplify the resulting expression to get $(f \circ g)(x)$.
Determine the domain of $f \circ g$ by identifying the values of $x$ that make the expression undefined. This involves considering the domains of both $f(x)$ and $g(x)$.

Step 1: Find $(f \circ g)(x)$

To find $(f \circ g)(x)$, we substitute $g(x) = \frac{5}{x}$ into $f(x) = \frac{4}{x + 8}$: \[ (f \circ g)(x) = f(g(x)) = f\left(\frac{5}{x}\right) = \frac{4}{\frac{5}{x} + 8} \] This simplifies to: \[ (f \circ g)(x) = \frac{4}{\frac{5 + 8x}{x}} = \frac{4x}{5 + 8x} \]

Step 2: Simplify the Expression

The expression $(f \circ g)(x)$ has been simplified to: \[ (f \circ g)(x) = \frac{4x}{8x + 5} \]

Step 3: Determine the Domain of $f \circ g$

The domain of $f \circ g$ is determined by the restrictions from both $f(x)$ and $g(x)$.

For $g(x) = \frac{5}{x}$, $x$ cannot be $0$.
For $f(x)$, the expression $x + 8$ must not be $0$, which does not impose any additional restrictions since $g(x)$ does not lead to $x + 8 = 0$.

Thus, the domain of $f \circ g$ is: \[ x \in \mathbb{R} \setminus \{0\} \]