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).