Questions: For f(x)=5/(x+7) and g(x)=8/x, find a. (f ∘ g)(x); b. the domain of f ∘ g a. (f ∘ g)(x)=5x/(7x+8) b. What is the domain of f ∘ g ? The domain is

Transcript text: For $f(x)=\frac{5}{x+7}$ and $g(x)=\frac{8}{x}$, find a. $(f \circ g)(x)$; b. the domain of $f \circ g$ a. $(f \circ g)(x)=\frac{5 x}{7 x+8}$ b. What is the domain of $f \circ g$ ? The domain is $\square$

Solution

Solution Steps

Solution Approach

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 expression to get the final form of $ (f \circ g)(x) $. For the domain of $ f \circ g $, determine the values of $ x $ for which both $ g(x) $ and $ f(g(x)) $ are defined. This involves ensuring that the denominators in both $ g(x) $ and the resulting expression from $ f(g(x)) $ are not zero.

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

To find $ (f \circ g)(x) $, we substitute $ g(x) = \frac{8}{x} $ into $ f(x) = \frac{5}{x + 7} $:

\[ (f \circ g)(x) = f(g(x)) = f\left(\frac{8}{x}\right) = \frac{5}{\frac{8}{x} + 7} \]

Simplifying this expression, we get:

\[ (f \circ g)(x) = \frac{5}{\frac{8 + 7x}{x}} = \frac{5x}{7x + 8} \]

Step 2: Determine the Domain of $ f \circ g $

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

The function $ g(x) = \frac{8}{x} $ is undefined when $ x = 0 $.
The expression $ f(g(x)) = \frac{5x}{7x + 8} $ is undefined when the denominator $ 7x + 8 = 0 $. Solving for $ x $:

\[ 7x + 8 = 0 \implies x = -\frac{8}{7} \]

Thus, the domain of $ f \circ g $ excludes $ x = 0 $ and $ x = -\frac{8}{7} $.

Step 3: Express the Domain in Interval Notation

The domain can be expressed in interval notation as:

\[ (-\infty, -\frac{8}{7}) \cup (-\frac{8}{7}, 0) \cup (0, \infty) \]