Question

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

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

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

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

f^−1(x)=3^x

Answer 1

To find the inverse of the function \( f(x) = \log_{1/3} x \), we need to express \( x \) in terms of \( y \), where \( y = f(x) \).

1. We start with the equation:
\[
y = \log_{1/3} x
\]

2. By the definition of logarithms, we can rewrite this in exponential form:
\[
x = (1/3)^y
\]

3. To find the inverse function, we solve for \( y \) in terms of \( x \):
\[
f^{-1}(x) = 3^x
\]

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