clarifying a thing about the inverse function $y=x^3$

Question

as the title states I need some clarification about inverse functions:

Take the function $y=x^3$ as an example.

If I want to inverse this function I first swap x and y:

I get: $x=y^3$

inorder to find the inverse of x I will have to take the third root of both sides:

this gives me : $x^\frac{1}{3}=y$ i.e $f^{-1}(x)=x^\frac{1}{3}$

However for the same function my book gives me the function

$f^{-1}(y)=y^\frac{1}{3}$

I presume they take the inverse of the function of x with respect to y.

i.e $x=f(y)$ which the same as $x=y^\frac{1}{3}$

Using the same steps as I did previously for findig the inverse $f^{-1}(y)$

I'll get $f^{-1}(y)=y^3$.

How do they get:

$f^{-1}(y)=y^\frac{1}{3}$??

Thank you in advance

Answer 1

I think you are confused. $f^{-1}(x)=x^{\frac 13}$ is exactly the same as $f^{-1}(y)=y^{\frac 13}$. Here $x,y$ are dummy variables.

Because of tradition, we express functions in term of the variable $x$, so we speak of $f(x)$ as well as $f^{-1}(x)$.

But for a function $f:A\mapsto B$ then

• for $f(x)$ we have $x\in A$
• while for $f^{-1}(x)$ we have $x\in B$.

You can see that we used the same name $x$ for different things, that's why the confusion arises.

Anyway, the alternative is to keep naming things $x$ for elements of $A$ and $y$ for elements of $B$ then it is more natural to speak of $f(x)$ and $f^{-1}(y)$.

Observing this convention then

Whenever $y=f(x)$ then $x=f^{-1}(y)$.

Since $y=x^3\iff x=y^{\frac 13}$ you get $f^{-1}(y)=y^{\frac 13}$

Now if you want to get back to the traditional way, you can replace the variable by anything you want : $f^{-1}(\square)=\square^{\frac 13}$

Put what you want into the square, in particular you can put $x$.

I don't know if it is clear, I have difficulties explaining this abstract concept.