Question

what is the inverse of the logarithmic function \\(f(x) = \log_{2}x\\)?

\\(f^{-1}(x) = x^2\\)
\\(f^{-1}(x) = 2^x\\)
\\(f^{-1}(x) = \log_{x}2\\)
\\(f^{-1}(x) = \frac{1}{\log_{2}x}\\)

Explanation:

Set up the inverse equation

Using the Inverse Functions knowledge point, we replace \(f(x)\) with \(y\) to find the inverse of the given function:

$$y = \log_{2}x$$

Swap the variables

To find the inverse function, we swap the independent variable \(x\) and the dependent variable \(y\):

$$x = \log_{2}y$$

Solve for y

We rewrite the logarithmic equation in its equivalent exponential form:

$$y = 2^x$$

Express as inverse function

We replace \(y\) with the inverse function notation \(f^{-1}(x)\):

$$f^{-1}(x) = 2^x$$

Answer:

• (A) \(f^{-1}(x) = x^2\)
• (B) \(f^{-1}(x) = 2^x\) (Correct answer)
• (C) \(f^{-1}(x) = \log_x 2\)
• (D) \(f^{-1}(x) = \frac{1}{\log_2 x}\)