Questions: For f(x)=4/(x+8) and g(x)=3/x, find a. (f ∘ g)(x); b. the domain of f ∘ g

Solution Steps

To solve the given problem, we need to find the composition of the functions \( f \) and \( g \), denoted as \( (f \circ g)(x) \). This involves substituting \( g(x) \) into \( f(x) \). After finding the composition, we will determine the domain of the composed function by identifying the values of \( x \) for which the function is defined.

To find the composition of the functions \( f(x) = \frac{4}{x + 8} \) and \( g(x) = \frac{3}{x} \), we substitute \( g(x) \) into \( f(x) \):

\[ (f \circ g)(x) = f(g(x)) = f\left(\frac{3}{x}\right) = \frac{4}{\frac{3}{x} + 8} \]

This simplifies to:

\[ (f \circ g)(x) = \frac{4}{\frac{3 + 8x}{x}} = \frac{4x}{8x + 3} \]

The simplified form of the composition is:

\[ (f \circ g)(x) = \frac{4x}{8x + 3} \]

The domain of \( (f \circ g)(x) \) is determined by the conditions under which both \( g(x) \) and \( f(g(x)) \) are defined.

1. \( g(x) = \frac{3}{x} \) is undefined when \( x = 0 \).
2. For \( f(g(x)) \) to be defined, we need \( g(x) + 8 \neq 0 \):

\[ \frac{3}{x} + 8 \neq 0 \implies \frac{3 + 8x}{x} \neq 0 \implies 3 + 8x \neq 0 \implies x \neq -\frac{3}{8} \]

Thus, the domain of \( (f \circ g)(x) \) is:

\[ x \neq 0 \quad \text{and} \quad x \neq -\frac{3}{8} \]

Final Answer