For the function f, of, x, equals, start fraction, x, divided by, 3, end fraction, plus, 8f(x)=

Question

3
x
​
+8, find f, to the power minus 1 , left parenthesis, x, right parenthesisf
−1
(x).
Answer
Attempt 1 out of 2
Multiple Choice Answers
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start fraction, left parenthesis, x, minus, 8, right parenthesis, divided by, 3, end fractionf
−1
(x)=
3
(x−8)
​

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 3, left parenthesis, x, minus, 8, right parenthesisf
−1
(x)=3(x−8)
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 3, left parenthesis, x, plus, 8, right parenthesisf
−1
(x)=3(x+8)
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 3, x, minus, 8f
−1
(x)=3x−8

Answer 1

To find the inverse function \( f^{-1}(x) \) for the given function \( f(x) = \frac{x}{3} + 8 \), we will follow these steps:

1. Replace \( f(x) \) with \( y \):

   \[ y = \frac{x}{3} + 8 \]

2. Swap \( x \) and \( y \):

   \[ x = \frac{y}{3} + 8 \]

3. Solve for \( y \):

   • First, subtract 8 from both sides:

   \[ x - 8 = \frac{y}{3} \]

   • Next, multiply both sides by 3:

   \[ 3(x - 8) = y \]

   • Thus,

   \[ y = 3(x - 8) \]

4. Express \( f^{-1}(x) \):

   \[ f^{-1}(x) = 3(x - 8) \]

Now let's review the multiple-choice answers:

1. \( f^{-1}(x) = \frac{x - 8}{3} \)
2. \( f^{-1}(x) = 3(x - 8) \) (Correct)
3. \( f^{-1}(x) = 3(x + 8) \)
4. \( f^{-1}(x) = 3x - 8 \)

The correct answer is:

\[ f^{-1}(x) = 3(x - 8) \]