Why can inverse functions be determined by manipulating equations?

Question

A common method for determining an inverse function $f^{-1}$ of a function $f$ is to write $f$ as an equation and solve for x. For example, if $f:=\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$, $x\mapsto x^2$, then $g:=\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$, $x\mapsto \sqrt{x}$ is an inverse function of $f$ because

$y=x^2$ iff $x=\sqrt{y}$.

(for another example see this post: https://math.stackexchange.com/a/2890915/493672)

Why does this method always work? The references I consulted seem to only state that this is how to do it, but not why it works. It's simply not evident to me. One problem I have is that functions are not equations, but in this case the authors write some function $f$ as an equation, treat it as an equation and get correct results anyway. It would be nice to have some justification for why inverse functions can be determined like this.

Edit: Fixed the range.

Answer 1

Let me answer on two levels.

First, if $y = f(x)$ for an invertible function $f$, then $x = f^{-1}(y)$. When you write out the equation $$y = f(x)$$ where $f(x)$ is some expression in $x$, then you solve this equation for $x$, the result will be in the form $$g(y) = x$$ or $x = g(y)$, for some other expression $g(y)$. But that expression defines a function $g$. And the very meaning "solve for $x$" is that the resultant equation $x = g(y)$ is equivalent to the original equation $y = f(x)$. But if we substitute for $x$ and $y$ in these equations from the other we get: $$x = g(f(x))\\y=f(g(y))$$ which is exactly the condition that $f$ and $g$ are inverse functions.

The second level is this. When you "solve $y = f(x)$ for $x$", what do you do? You make a series of changes to the equation to get an equivalent equation. The key rule is that if you do something to one side of the equation, you must do the same thing to the other side of the equation. But those "things" that you do can be thought of as applying some function to both sides:

Add something? $$g(x,y) = f(x,y) \longrightarrow g(x,y) + a = f(x,y) + a$$ is another way of writing $$g(x,y) = f(x,y) \longrightarrow h(g(x,y))= h(f(x,y))$$ where $h(t) = t + a$. Multiplying is the same, where $h(t) = at$. Taking the square is $h(t) = t^2$, etc.

So you start with $y = f(x)$, and step-by-step apply a sequence of functions: $$y = f(x)\\h_1(y) = h_1(f(x)) = h_1\circ f(x)\\h_2\circ h_1(y) = h_2\circ h_1\circ f(x)\\\vdots\\h_n\circ \dots \circ h_2\circ h_1(y) = h_n\circ \dots \circ h_2\circ h_1\circ f(x)$$

But there is a point to this. If you do it right, in the end $h_n\circ \dots \circ h_2\circ h_1\circ f(x) = x$, so you end up with $$h_n\circ \dots \circ h_2\circ h_1(y) = x$$ Letting $g = h_n\circ \dots \circ h_2\circ h_1$, this is $$x = g(y)$$ As before, this means $g$ is the inverse of $f$.

Answer 2

@PaulSinclair's answer is great for the specific case you're looking at, but I think it's worthwhile to address the "why" more generally.

Mathematics is a language, in the sense that it builds up information through both syntax and semantics; and distinguishes true from false through propositions (relative to a collection of axioms). Syntax is the rules of what makes "a correct sentence". Semantics is the informational meaning of the sentence. Propositions are truth statements: sentences that can be evaluated as either true or false (or undecidable, in rare cases).

For example, in vernacular language:

• "Ate dog boy" is syntactically invalid.

• "Green ideas sleep furiously." (Chomsky) is syntatically valid, but semantically invalid. Do note that syntactic validity in no way implies semantic validity.

• "Am I hungry ?" is both syntactically and semantically valid, but not a proposition.

• "Santa Claus exists." is syntactically, semantically, and propositionally valid, but propositionally false (at least in this realm of existence; and with Santa Claus defined as the mythical flying man, rather than the mall version).

• "Some birds can fly" is syntactically, semantically, and propositionally valid, and propositionally true.

As for mathematical language, similar thing can be said:

• "$x \; 2 \to =$" is syntactically invalid

• "$x \in \mathbb{R}, \frac{x}{0}$" is syntactically valid, but semantically invalid

• "Is the theorem $2 + 2 = 5$ true ?" is syntactically valid, semantically valid, but propositionally invalid.

• "$2 + 2 = 5$" is syntactically, semantically, propositionally valid, but propositionally false.

• "$2 + 2 = 4$" is syntactically, semantically, and propositionally valid and propositionally true.

So how is this relevant to your problem specifically ? Consider the following:

• "The boy is petting the dog."

• "The dog is being pet by the boy."

and

• $A = \pi r^2$, the area of a circle is the radius squared times the constant $3.14 \dots$

• $r = \sqrt{\frac{A}{\pi}}$, the radius of a circle is the square root of its area-divided-by-$3.14 \dots$

We have:

$$\text{"The boy is petting the dog."} \Leftrightarrow \text{"The dog is being pet by the boy."}$$

and

$$A = \pi r^2 \Leftrightarrow r = \sqrt{\frac{A}{\pi}}$$

Both pairs of statements, while syntactically distinct (active form vs passive form), are semantically identical (equivalent). This is what logical equivalence (your "iff", and more generally, isomorphisms) does for you in mathematics: it allows you to know when distinct mathematical representations "mean" (or "refer to") the same thing.

Finding the reciprocal of a bijective function is precisely this sort of passage "from the active form to the passive form". That's "why" it works.