What is the difference between inverse and reciprocal?

Question

Okay soo, i always thought that the inverse of $y=e^x$ is $\frac{1}{e^x}$ but its not??? It is y= $\log_e(x)$?

I swear these terms have been used interchangeably. So can someone please explain the difference between reciprocal and inverse?

Thanks.

Answer 1

The reciprocal is what you would multiply by in order to obtain $1$. So for the fraction $\frac{1}{2}$, this would be $\frac{2}{1}$. For the fraction $\frac{3}{4}$, this would be $\frac{4}{3}$. For any $x$, the reciprocal of $e^x$ would be $\frac{1}{e^x}$, because observe $e^x \cdot \dfrac{1}{e^x}= 1$.

However, the inverse is what you compose with to obtain the input value. So for instance, if $f(x)= e^x$, the inverse is $g(x)= \ln x$. Because then $g(f(2))=2$, $g(f(16))=16$, $g(f(-10))=-10$, etc. But notice this is not the case with $1/e^x$. For instance, if $x=3$, then $e^3 \cdot \frac{1}{e^3}=1 \neq 3$.

The difference is what you want out of the 'operation'. In one case, reciprocals, you want to obtain $1$ from a product. In the case of inverses, you want to 'undo' a function and obtain the input value. Of course, all of the above discussion glosses over that not all functions have inverses (or perhaps only a left/right inverse) and reciprocals for functions are not always defined (for instance, whenever the function take on $0$ as a value).

Answer 2

Since you are asking under the term "calculus", my guess is that the reason why you think that you have heard "reciprocal" and "inverse function" being used interchangeably is that, in calculus, the derivative of the inverse function is the reciprocal of the derivative of the main function. Either that, or your teacher is getting their terms confused (it happens, sorry!).

That is, if $y = e^x$, the "inverse function" is the function that reverses that process. I.e., in the original equation, if you put $x = 0$ into the input, you get $y = 1$ coming out. To reverse this, we want a function where putting $x = 1$ to the input will get $y = 0$ coming out, and similar equivalences for all outputs of the original function.

The way you find the inverse function is simply swapping $x$ and $y$ in the original equation, and solving for $y$ again. So, if we have $y = e^x$, to find the inverse function we will just swap $x$ and $y$. This gives us $x = e^y$. But we want to solve for $y$. So, we will use the natural log. $\ln(x) = y$.

Now, let's look at the derivatives of both the original and the inverse function. The derivative of the first function is $\frac{dy}{dx} = e^x$. The derivative of the inverse function is $\frac{dy}{dx} = \frac{1}{x}$. These might not seem to be reciprocals, until you realize that $x$ and $y$ mean something different in each function. If we translate the second derivative to use the same variable names as the first, then the derivative is actually $\frac{1}{y}$, and, since $y = e^x$, this is $\frac{1}{e^x}$, which is the reciprocal of the first derivative.

This is actually fairly straightforward from the perspective of differentials. The derivative of the function is $\frac{dy}{dx}$, and the derivative of the inverse function is $\frac{dx}{dy}$, which is just the reciprocal of the main derivative.