Question

Find the inverse function of the function f, of, x, equals, one third, x, plus, 1f(x)=
3
1
​
x+1.
Answer
Attempt 1 out of 2
Multiple Choice Answers
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 3, x, minus, 3f
−1
(x)=3x−3
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, one third, x, minus, 3f
−1
(x)=
3
1
​
x−3
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, one third, x, minus, 1f
−1
(x)=
3
1
​
x−1
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 3, x, minus, 1f
−1
(x)=3x−1

Answer 1

To find the inverse function of \( f(x) = \frac{1}{3}x + 1 \), we follow these steps:

1. **Replace \( f(x) \) with \( y \)**:
\[
y = \frac{1}{3}x + 1
\]

2. **Solve for \( x \)** in terms of \( y \)**:
- First, subtract 1 from both sides:
\[
y - 1 = \frac{1}{3}x
\]
- Then, multiply both sides by 3:
\[
3(y - 1) = x
\]

3. **Express the inverse function**:
- We can rewrite this as:
\[
x = 3y - 3
\]
- Now, swap \( x \) and \( y \):
\[
f^{-1}(x) = 3x - 3
\]

Thus, the inverse function \( f^{-1}(x) = 3x - 3 \).

**Correct Answer**:
\[ f^{-1}(x) = 3x - 3 \]

This corresponds to the answer choice: **f, to the power minus 1, left parenthesis, x, right parenthesis, equals, 3, x, minus, 3**.