How is $1/f$ called?

Question

For numbers, one calls $\frac 1x$ the inverse of $x$

However, the inverse of a function $f$ is $f^{-1}$, i.e. the function which composed with $f$ gives the identity function

Is anyone aware of a name for the function $1/f$?

thanks in advance!

Answer 1

The term "inverse" is related to an operation. So if you define the operation of multiplying an object (say the function) $f:\mathbb R \to \mathbb R$ by (say the real number) $x$, i.e.

$$M f(t)= x \, f(t).$$

Then, the inverse of the operation $M$ is (usually) denoted by $M^{-1}$, and defined as

$$M^{-1} f(t) = \frac{1}{x} f(t).$$

Then inverse here means that $M M^{-1} f(f) = M^{-1} M f(t)=f(t)$. i.e. $M M^{-1} = M^{-1} M =I$ where $I$ is the identity operation, that is $I f(t) =f(t)$.

Therefore, the inverse is for operations, and the reciprocal (1/x is the reciprocal of x) is the inverse of a particular operation, which is "multiplication."

Note: Division is multiplication by the reciprocal.

Note that