Answer :
To find the inverse of the logarithmic function [tex]\( f(x) = \log_2 x \)[/tex], we need to understand the concept of inverse functions. The inverse function essentially "reverses" the operation of the original function.
1. Define the function and its purpose:
- The function [tex]\( f(x) = \log_2 x \)[/tex] means finding a power [tex]\( y \)[/tex] (output) such that [tex]\( 2^y = x \)[/tex] (input).
2. Rewrite the function in exponentiation form:
- If [tex]\( y = \log_2 x \)[/tex], this implies that [tex]\( 2^y = x \)[/tex].
3. Solve for the input [tex]\( x \)[/tex] in terms of the output [tex]\( y \)[/tex]:
- To find the inverse function [tex]\( f^{-1}(x) \)[/tex], we need to swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex], and then solve for the new output in terms of the new input.
- Swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex] gives [tex]\( x = \log_2 y \)[/tex], which means [tex]\( 2^x = y \)[/tex].
4. Writing the inverse function:
- Since [tex]\( 2^x = y \)[/tex], the inverse function [tex]\( f^{-1}(x) \)[/tex] is [tex]\( 2^x \)[/tex].
Hence, the inverse of the function [tex]\( f(x) = \log_2 x \)[/tex] is:
[tex]\[ f^{-1}(x) = 2^x \][/tex]
Therefore, the correct answer is:
[tex]\[ f^{-1}(x) = 2^x \][/tex]
1. Define the function and its purpose:
- The function [tex]\( f(x) = \log_2 x \)[/tex] means finding a power [tex]\( y \)[/tex] (output) such that [tex]\( 2^y = x \)[/tex] (input).
2. Rewrite the function in exponentiation form:
- If [tex]\( y = \log_2 x \)[/tex], this implies that [tex]\( 2^y = x \)[/tex].
3. Solve for the input [tex]\( x \)[/tex] in terms of the output [tex]\( y \)[/tex]:
- To find the inverse function [tex]\( f^{-1}(x) \)[/tex], we need to swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex], and then solve for the new output in terms of the new input.
- Swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex] gives [tex]\( x = \log_2 y \)[/tex], which means [tex]\( 2^x = y \)[/tex].
4. Writing the inverse function:
- Since [tex]\( 2^x = y \)[/tex], the inverse function [tex]\( f^{-1}(x) \)[/tex] is [tex]\( 2^x \)[/tex].
Hence, the inverse of the function [tex]\( f(x) = \log_2 x \)[/tex] is:
[tex]\[ f^{-1}(x) = 2^x \][/tex]
Therefore, the correct answer is:
[tex]\[ f^{-1}(x) = 2^x \][/tex]
