Edwards Malone
11/06/2023 · High School
Find \( (f \circ g)(x) \) and \( (g \circ f)(x) \) and the domain of each. \( f(x)=6 x-4, g(x)=\frac{x+4}{6} \)
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Tutor-Verified Answer
Step-by-step Solution
To find \( (f \circ g)(x) \) and \( (g \circ f)(x) \), we will first compute each composition of functions and then determine their domains.
### Step 1: Compute \( (f \circ g)(x) \)
The composition \( (f \circ g)(x) \) means we will substitute \( g(x) \) into \( f(x) \).
1. First, find \( g(x) \):
\[
g(x) = \frac{x + 4}{6}
\]
2. Now substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f\left(\frac{x + 4}{6}\right) = 6\left(\frac{x + 4}{6}\right) - 4
\]
3. Simplify:
\[
f(g(x)) = (x + 4) - 4 = x
\]
Thus, we have:
\[
(f \circ g)(x) = x
\]
### Step 2: Determine the domain of \( (f \circ g)(x) \)
The function \( g(x) = \frac{x + 4}{6} \) is defined for all real numbers \( x \). Therefore, the domain of \( (f \circ g)(x) \) is:
\[
\text{Domain of } (f \circ g) = \mathbb{R}
\]
### Step 3: Compute \( (g \circ f)(x) \)
The composition \( (g \circ f)(x) \) means we will substitute \( f(x) \) into \( g(x) \).
1. First, find \( f(x) \):
\[
f(x) = 6x - 4
\]
2. Now substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(6x - 4) = \frac{(6x - 4) + 4}{6}
\]
3. Simplify:
\[
g(f(x)) = \frac{6x}{6} = x
\]
Thus, we have:
\[
(g \circ f)(x) = x
\]
### Step 4: Determine the domain of \( (g \circ f)(x) \)
The function \( f(x) = 6x - 4 \) is defined for all real numbers \( x \). Therefore, the domain of \( (g \circ f)(x) \) is:
\[
\text{Domain of } (g \circ f) = \mathbb{R}
\]
### Summary
- \( (f \circ g)(x) = x \) with domain \( \mathbb{R} \)
- \( (g \circ f)(x) = x \) with domain \( \mathbb{R} \)
Quick Answer
\( (f \circ g)(x) = x \), domain \( \mathbb{R} \)
\( (g \circ f)(x) = x \), domain \( \mathbb{R} \)
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