Is it correct to say the reciprocal of a function is $1$ over that function, and the inverse is the function reflected over the line $y = x$?

Question

I am wondering if you can use the reflection over $y = x$ to explain the inverse of a function vs a reciprocal. Not sure if this is true for all functions however. I understand the reciprocal is what you need to multiply a function (or anything with) to get back to $1/1$, but it's hard to differentiate that with the inverse.

Answer 1

In line with the comments above:

Reflecting across $y=x$ can be a useful aid when studying real-valued functions of one variable.

Inverse from the function point of view and (multiplicative) inverse = reciprocal are related for $f(x)=1/x$ but not the same thing. $f(x)=1/x$ is its own inverse, however $g(x)=x$ and $h(x)=1/x$ are multiplicative inverses of each other.

The inverse of a function is another function that "gets you back where you started", meaning for a set $A$ and function $f: A\rightarrow A$ we would call $g$ the inverse of $f$ if $f(g(a))=g(f(a))$ for all $a\in A$, and write $f\circ g=g\circ f=1_A$ where $1_A$ is the identity function on $A$.

I've left out various details to keep this relatively simple.

Where multiplication and division make sense, then $f(a) = 1/a$ (reciprocal) is its own inverse in the above sense.

Which is probably what leads to the notation of $f^{-1}$ to mean the inverse function and generally not the reciprocal.

This usually arises for the first time when first studying trig functions and making clear that $\sin^{-1}x$ is most definitely the inverse and not the reciprocal.