Simple inverse function question - $f(X) = \frac{2}{X}$.

Question

I have been studying functions and how to find the inverse of a function which so far I have found straight forward. However, I have had some difficulty while trying to find the inverse of function $f(X) = \frac{2}X$. I have the answer sheet to the question, which says the answer is "$f^{-1}(x) = \frac{2}x$".

I took the function to mean that the input maps to $2$ divided by $X$, so I thought the inverse would be $2$ multiplied by $X$, or "$f^{-1}(x) = 2 x$"

Am I making a mistake with the notation? I have found much more complicated functions easier to find the inverse of, I'm not sure what rookie mistake I'm making here.

Answer 1

Your error is understandable. Note however that, with $f(x) = \frac2x$ we have:

$$\begin{cases}f(1) = 2\\f(2) = 1\\f(4) = \frac12\\\cdots\end{cases}$$

so the inverse can't certainly be $f^{-1}(x) = 2x$ given that it will only work for one of the examples above. To find the inverse, just solve the equation

$$f(x) = \frac2x$$ for $x$. One usually writes that as $y = \frac2x$ and solves for $x$. The equation writes the image as a function of the object. Since you want to find the object as a function of the image, solve the equation for the object:

$$y = \frac2x \iff x = \frac2y \implies f^{-1}(x) = \frac2x$$

Answer 2

Set $y=f(x)=\frac{2}{x}$, and solve for $x$. We multiply both sides by $x$ to get $xy=2$, and divide both sides by $y$ to get $x=\frac{2}{y}$. Hence $f^{-1}(y)=\frac{2}{y}$, and (by changing variables) $f^{-1}(x)=\frac{2}{x}$.

Time to check our work. We calculate $$f^{-1}(f(x))=\frac{2}{f(x)}=\frac{2}{\frac{2}{x}}=\frac{2}{1}\frac{x}{2}=x$$

This proves that $f^{-1}(x)=\frac{2}{x}$ is indeed the correct inverse.

Answer 3

$y=2/x$ iff $xy=2$ iff $x=2/y$. Hence $f^{-1}(x)=2/x$