Question
Watch the video and then solve the problem given below. Click here to watch the video. Find \( f(g(x)) \) and \( g(f(x)) \) and determine whether the pair of functions \( f \) and \( g \) are inverses of each other. \( f(x)=3 x+1 \) and \( g(x)=\frac{x-1}{3} \) a. \( f(g(x))=\square \) (Simplify your answer.)
Ask by Daniels Schmidt. in the United States
Mar 17,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f(g(x)) = x \)
Solution
**Step 1: Compute \( f(g(x)) \)**
Given:
\[
f(x) = 3x + 1 \quad \text{and} \quad g(x) = \frac{x-1}{3}
\]
Substitute \( g(x) \) into \( f \):
\[
f(g(x)) = f\left(\frac{x-1}{3}\right)
\]
Using the definition of \( f \):
\[
f\left(\frac{x-1}{3}\right) = 3\left(\frac{x-1}{3}\right) + 1
\]
Simplify the expression:
\[
3\left(\frac{x-1}{3}\right) = x - 1
\]
\[
x - 1 + 1 = x
\]
Thus, the simplified result is:
\[
f(g(x)) = x
\]
**Step 2: Compute \( g(f(x)) \)**
Substitute \( f(x) \) into \( g \):
\[
g(f(x)) = g(3x+1)
\]
Using the definition of \( g \):
\[
g(3x+1) = \frac{(3x+1)-1}{3}
\]
Simplify the expression:
\[
(3x+1)-1 = 3x
\]
\[
\frac{3x}{3} = x
\]
Thus, the simplified result is:
\[
g(f(x)) = x
\]
**Conclusion:**
Since both compositions yield:
\[
f(g(x)) = x \quad \text{and} \quad g(f(x)) = x
\]
the functions \( f \) and \( g \) are indeed inverses of each other.
**Final Answers:**
a. \( f(g(x)) = x \)
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Simplify this solution Mind Expander
To find \( f(g(x)) \), we start with \( g(x) = \frac{x-1}{3} \). Now, we substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\frac{x-1}{3}\right) = 3\left(\frac{x-1}{3}\right) + 1 = (x-1) + 1 = x. \] So, \( f(g(x)) = x \). Now, let's calculate \( g(f(x)) \) using \( f(x) = 3x + 1 \). \[ g(f(x)) = g(3x + 1) = \frac{(3x + 1) - 1}{3} = \frac{3x}{3} = x. \] So, \( g(f(x)) = x \). Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f \) and \( g \) are indeed inverses of each other!
