# Inverse & preimage

1) Consider the example y=f(x)=x3
My statistics textbook say x=f -1(y)=y1/3 is the inverse of f
On the other hand, my calculus textbook says y=x1/3 is the inverse of f
So I am confused...it looks like the idea of inverse is used inconsistently. (When you plot both functions on the xy-plane, you will certainly see two different graphs.)
Which one is the correct one according to the precise definition of inverse?

2) I don't get the difference between the "inverse" of f and the "inverse image" or "premiage". Can somebody explain?

Thank you!

1) Consider the example y=f(x)=x3
My statistics textbook say x=f -1(y)=y1/3 is the inverse of f
On the other hand, my calculus textbook says y=x1/3 is the inverse of f
So I am confused...it looks like the idea of inverse is used inconsistently. (When you plot both functions on the xy-plane, you will certainly see two different graphs.)
Which one is the correct one according to the precise definition of inverse?

Note in this example that f-1 is written as a function of y instead of x. That's where your confusion is coming from. The inverse function could (and for clarity's sake, probably should) be written as $$f^{-1}(x)=x^{1/3}$$.

2) I don't get the difference between the "inverse" of f and the "inverse image" or "premiage". Can somebody explain?

Suppose f is a function. Another function f-1 is called the inverse of f if $$f(f^-1(x)) = f^-1(f(x)) = x$$ for all x. In other words, an inverse function is just a function with a special relation to another.

The preimage of a point under a function is a the set of points which map to that point. In other words preimage(p) = {x such that f(x) = p}. So the preimage of a point is a set.

HallsofIvy
Homework Helper
By the way, even if a function, f, does not have an inverse, we can still define the "inverse image", f-1(A). For example, if f(x)= x2, there is no "f-1(x)" because f is not "one-to-one"; since f(2)= 4 and f(-2)= 4, which would be f-1(4)?

But if B= [0, 1], we can still have f-1(B)= [-1, 1] since, for any x in [-1, 1], f(x)= x2 is in [0,1].

I once made a fool of myself, presenting a proof in a graduate class, by forgetting that! I was to prove a statement about inverse images and did it assuming the function f must have an inverse function.

By the way, even if a function, f, does not have an inverse, we can still define the "inverse image", f-1(A).

Ah yes! This is an important point!

We can also "map" functions over sets. So if A is a set of numbers and f is a function, then we define f(A) = {f(x) for each x in A}.

This gives us cute little properties like

$$x \in f(A) \Leftrightarrow f^{-1}(x) \in A$$

Thanks! Now I have an idea of the difference between inverse & premiage.

Back to 1) y=f(x)=x^3
Is it even correct to say that x=f -1(y)=y^(1/3) is the inverse of f ? My statsitics textbook is keep doing the same thing again and again...but then there would be inconsistency...x=y^(1/3) and y=x^(1/3) do not give the same graph when you graph them on the xy-plane.

HallsofIvy
When you have a function defined as $$f(x) = x^3$$, the function itself is named "f". You would say "f is a cubic function". However, it is very common to confuse "f" with "f(x)" and say that "f(x) is a cubic function". Technically f(x) means "f evaluated at x" or "f with the argument x supplied to it" or something, but in practice, it's usually clear from context what you mean.
A related consequence is that the variable name DOES NOT MATTER. It is arbitrary. If $$f(x) = x^3$$, then it is just as true to say $$f(y) = y^3$$. Just like how $$\Sigma_{i=0}^\infty \frac{1}{2^i}$$ is the same as $$\Sigma_{k=0}^\infty \frac{1}{2^k}$$. j and k are just dummy variables. Variables used in the definition of functions are the same.