Questions: Given f(x)=1/(x-4) and g(x)=4/(x-3), find the domain of f(g(x)) Domain (in interval notation):

Transcript text: Given $f(x)=\frac{1}{x-4}$ and $g(x)=\frac{4}{x-3}$, find the domain of $f(g(x))$ Domain (in interval notation): $\square$

Solution

Solution Steps

To find the domain of the composite function $ f(g(x)) $, we need to consider the domains of both $ f(x) $ and $ g(x) $. The function $ g(x) $ is defined for all $ x \neq 3 $ because the denominator cannot be zero. The function $ f(x) $ is defined for all $ x \neq 4 $. Therefore, for $ f(g(x)) $ to be defined, $ g(x) $ must not equal 4. We solve $ g(x) = 4 $ to find the values of $ x $ that make $ f(g(x)) $ undefined. Finally, we combine these restrictions to find the domain of $ f(g(x)) $.

Step 1: Determine the Domain of $ g(x) $

The function $ g(x) = \frac{4}{x - 3} $ is defined for all $ x $ except where the denominator is zero. Thus, the domain of $ g(x) $ is: \[ x \neq 3 \]

Step 2: Find Values that Make $ f(g(x)) $ Undefined

Next, we need to find when $ f(g(x)) $ is undefined. The function $ f(x) = \frac{1}{x - 4} $ is undefined when its argument equals 4. We set $ g(x) = 4 $: \[ \frac{4}{x - 3} = 4 \] Solving this equation, we find: \[ 4 = 4(x - 3) \implies 4 = 4x - 12 \implies 4x = 16 \implies x = 4 \] Thus, $ g(x) = 4 $ when $ x = 4 $.

Step 3: Combine Domain Restrictions

The domain of $ f(g(x)) $ must exclude both the point where $ g(x) $ is undefined and the point where $ g(x) = 4 $. Therefore, we exclude $ x = 3 $ and $ x = 4 $ from the domain. The resulting domain in interval notation is: \[ (-\infty, 3) \cup (3, 4) \cup (4, \infty) \]