Question

Which function is the inverse of f(x)=log1/3 x
?(1 point)
Responses

f−1(x)=x13
f superscript negative 1 baseline left parenthesis x right parenthesis equals x superscript Start Fraction 1 over 3 End Fraction baseline

f−1(x)=3x
f superscript negative 1 baseline left parenthesis x right parenthesis equals 3 superscript x baseline

f−1(x)=x3
f superscript negative 1 baseline left parenthesis x right parenthesis equals x superscript 3 baseline

f−1(x)=(13)x

Answer 1

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

1. Rewrite the function in exponential form:
\[
y = \log_{1/3} x \implies x = (1/3)^y
\]

2. Now, solve for \( y \):
\[
x = \frac{1}{3^y} \implies 3^y = \frac{1}{x} \implies y = -\log_3(x)
\]
However, we can also recognize that to transition from a logarithmic form to the inverse function, we can simply rewrite it as:
\[
y = \log_{1/3} x \implies x = 3^{-y}
\]

3. Switching \( x \) and \( y \):
\[
f^{-1}(x) = 3^{-x}
\]

Given these transformations, the correct choice from the provided options is:
\[
f^{-1}(x) = 3^x
\]

Therefore, the correct response is:
\[
f^{-1}(x) = 3^x
\]