Answer: The correct option is (D) [tex]f(x)=2x.[/tex]
Step-by-step explanation: We are given to select the correct function for which f(x) is not equal to [tex]f^{-1}(x).[/tex]
Option (A) :
Here,
[tex]f(x)=2-x.[/tex]
Let f(x) = y, so [tex]x=f^{-1}(y).[/tex]
Therefore,
[tex]f(x)=2-x\\\\\Rightarrow y=2-f^{-1}(y)\\\\\Rightarrow f^{-1}(y)=2-y\\\\\Rightarrow f^{-1}(x)=2-x.[/tex]
So, the function is equal to its inverse.
This option is incorrect.
Option (B) :
Here,
[tex]f(x)=\dfrac{2}{x}.[/tex]
Therefore,
[tex]f(x)=\dfrac{2}{x}\\\\\Rightarrow y=\dfrac{2}{f^{-1}(y)}\\\\\Rightarrow f^{-1}(y)=\dfrac{2}{y}\\\\\Rightarrow f^{-1}(x)=\dfrac{2}{x}.[/tex]
So, the function is equal to its inverse.
This option is incorrect.
Option (C) :
Here,
[tex]f(x)=-x.[/tex]
Therefore,
[tex]f(x)=-x\\\\\Rightarrow y=-f^{-1}(y)\\\\\Rightarrow f^{-1}(y)=-y\\\\\Rightarrow f^{-1}(x)=-x.[/tex]
So, the function is equal to its inverse.
This option is incorrect.
Option (D) :
Here,
[tex]f(x)=2x.[/tex]
Therefore,
[tex]f(x)=2x\\\\\Rightarrow y=2f^{-1}(y)\\\\\Rightarrow f^{-1}(y)=\dfrac{y}{2}\\\\\Rightarrow f^{-1}(x)=\dfrac{x}{2}.[/tex]
So, the function is NOT equal to its inverse.
This option is correct.
Thus, (D) is the correct option.
