If `f(x)=(1)/(3)x + 3`, then `f^(-1)(x)=`

Options

A. `-(1)/(3)x-3`
B. `-3x+(1)/(3)`
C. `3x+(1)/(3)`
D. `3x-9`

Explanation

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

### Step 1: Set \( f(x) \) equal to \( y \)
We start by letting \( y = f(x) \):
\[ y = \frac{1}{3}x + 3 \]

### Step 2: Solve for \( x \) in terms of \( y \)
Next, we need to rearrange the equation to express \( x \) in terms of \( y \). First, we will isolate the term involving \( x \):
\[ y - 3 = \frac{1}{3}x \]

### Step 3: Eliminate the fraction
To eliminate the fraction, multiply both sides by 3:
\[ 3(y - 3) = x \]
This simplifies to:
\[ x = 3y - 9 \]

### Step 4: Write the inverse function
Now that we have \( x \) in terms of \( y \), we can express the inverse function. We replace \( y \) with \( x \) and \( x \) with \( f^{-1}(x) \):
\[ f^{-1}(x) = 3x - 9 \]

### Final Answer
Thus, the inverse function is:
\[ f^{-1}(x) = 3x - 9 \]

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