If `f:RtoR^(+)` such that `f(x)=(1//3)^(x)`, the what is the value of `f^(-1)(x)`?

Options

A. `(1//3)^(x)`
B. `3^(x)`
C. `log_(1//3)X`
D. `log_(x)(1//3)`

Explanation

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

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

### Step 2: Solve for \( x \) in terms of \( y \)
To find the inverse, we need to express \( x \) in terms of \( y \). We can rewrite the equation:
\[ y = \left(\frac{1}{3}\right)^x \]
Taking the logarithm base \( \frac{1}{3} \) on both sides:
\[ \log_{\frac{1}{3}}(y) = \log_{\frac{1}{3}}\left(\left(\frac{1}{3}\right)^x\right) \]
Using the property of logarithms that states \( \log_b(a^c) = c \cdot \log_b(a) \):
\[ \log_{\frac{1}{3}}(y) = x \cdot \log_{\frac{1}{3}}\left(\frac{1}{3}\right) \]

### Step 3: Simplify the logarithm
Since \( \log_{\frac{1}{3}}\left(\frac{1}{3}\right) = 1 \):
\[ \log_{\frac{1}{3}}(y) = x \cdot 1 \]
Thus, we have:
\[ x = \log_{\frac{1}{3}}(y) \]

### Step 4: Express the inverse function
Now that we have \( x \) in terms of \( y \), we can express the inverse function \( f^{-1}(x) \):
\[ f^{-1}(x) = \log_{\frac{1}{3}}(x) \]

### Final Answer
Therefore, the value of \( f^{-1}(x) \) is:
\[ f^{-1}(x) = \log_{\frac{1}{3}}(x) \]

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