Find the inverse function of the function f, of, x, equals, one third, x, plus, 1f(x)=

Question

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.