Question

Which function is the inverse of f of x equals 3 left-parenthesis 2 right-parenthesis Superscript x Baseline?

Answer options with 5 options
A.
f inverse of x equals StartFraction 1 over 3 left-parenthesis 2 right-parenthesis Superscript x Baseline EndFraction

B.
f inverse of x equals StartFraction 1 over 3 EndFraction left-parenthesis StartFraction 1 over 2 EndFraction right-parenthesis Superscript negative x Baseline

C.
f inverse of x equals StartFraction Log base 2 of left-parenthesis x right-parenthesis EndLog over 3 EndFraction

D.
f inverse of x equals Log base 2 of left-parenthesis StartFraction x over 3 EndFraction right-parenthesis EndLog

E.
f inverse of x equals Log base 3 of left-parenthesis StartFraction x over 2 EndFraction right-parenthesis EndLog

Answer 1

To find the inverse of the function \( f(x) = 3(2^x) \), we'll follow these steps:

1. Replace \( f(x) \) with \( y \):
\[
y = 3(2^x)
\]

2. Swap \( x \) and \( y \):
\[
x = 3(2^y)
\]

3. Solve for \( y \):
\[
\frac{x}{3} = 2^y
\]

4. Take the logarithm base 2 of both sides:
\[
y = \log_2\left(\frac{x}{3}\right)
\]

Now we have found the inverse function:
\[
f^{-1}(x) = \log_2\left(\frac{x}{3}\right)
\]

Looking at the answer options, the correct choice is:

**D. \( f^{-1}(x) = \log_2\left(\frac{x}{3}\right) \)**.