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I am stuck with this question,

Let $A=B=C=\mathbb{R}$ and consider the functions $f\colon A\to B$ and $g\colon B\to C$ defined by $f(a)=2a+1$, $g(b)=b/3$. Verify Theorem 3(b): $(g\circ f)^{-1}=f^{-1}\circ g^{-1}.$

I have calculated $f^{-1}$, $g^{-1}$, and their composition, but how do I find the inverse of $(g\circ f)$?

Here is how I have done so far,

$$\begin{align*} \text{Let}\qquad\qquad b &= f(a)\\ a&= f^{-1}(b)\\ &{ }\\ b&=f(a)\\ b&=2a+1\\ \frac{b-1}{2} &= a\\ a &= \frac{b-1}{2} \end{align*}$$ But $a=f^{-1}(b)$, $$f^{-1}(b) = \frac{b-1}{2}.$$

${}$

$$\begin{align*} \text{Let}\qquad\qquad a&=g(b)\\ b&= g^{-1}(a)\\ a&= g(b)\\ a &= b/3\\ b &= 3a\\ g^{-1}(a) &= 3a\qquad(\text{because }b=g^{-1}(a) \end{align*}$$

${}$

$$\begin{align*} f^{-1}\circ g^{-1} &= ?\\ &= f^{-1}\Bigl( g^{-1}(a)\Bigr)\\ &= f^{-1}(3a)\\ f^{-1}\circ g^{-1} &= \frac{3a-1}{2} \end{align*}$$

$$\begin{align*} g\circ f&= g\bigl(f(a)\bigr)\\ &= g(2a+1)\\ g\circ f &= \frac{2a+1}{3}\\ (g\circ f)^{-1} &= ?\\ \text{Let}\qquad\qquad &b=g\circ f \end{align*}$$

EDIT:

Thanks for the answers, I followed the suggestions and came up with the answer,Now I have two questions,

1. The answers do match but the arguments are different. Is that ok?
2. Is $(g\circ f)$ same as $(g\circ f(a))$?

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3 Answers

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$f,g$ are the functions defined in the question.

We have

$$\begin{eqnarray*} b &=&f(a)=2a+1, \end{eqnarray*}$$

or equivalently, by definition of the inverse function $f^{-1}$

$$\begin{eqnarray*} &a=\frac{b-1}{2}=f^{-1}(b).\tag{A} \end{eqnarray*}$$

Since

$$\begin{eqnarray*} c &=&g(b)=\frac{b}{3}, \end{eqnarray*}$$

or equivalently, by definition of the inverse function $g^{-1}$

$$\begin{eqnarray*} b=3c=g^{-1}(c),\tag{B} \end{eqnarray*}$$

after combining $(A)$ and $(B)$, we get

$$a=\frac{3c-1}{2}=(f^{-1}\circ g^{-1})(c).\tag{1}$$

On the other hand

$$c=(g\circ f)(a)=g(f(a))=g(2a+1)=\frac{2a+1}{3}.\tag{2}$$

Hence, by definition, the value at $c$ of the inverse function $(g\circ f)^{-1}$, is

$$a=\frac{3c-1}{2}=(g\circ f)^{-1}(c).\tag{3}$$

From $(1)$ and $(3)$ we conclude that for these functions $f,g$ and their inverses $f^{-1},g^{-1}$ the following identity holds:

$$(f^{-1}\circ g^{-1})(c)=(g\circ f)^{-1}(c).\tag{4}$$

Notation's note: $(f^{-1}\circ g^{-1})(c)=f^{-1}(g^{-1}(c))$.

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What you've done so far is to compute $f^{-1}$ and $g^{-1}$, and $f^{-1}\circ g^{-1}$. Now you want to try to find $(g\circ f)^{-1}$ directly, and compare that to what you've computed (in order to verify the formula).

So, you've figured out that $(g\circ f)(a) = \frac{2a+1}{3}$. How do we figure out $(g\circ f)^{-1}$?

Exactly the same way we figure out the inverse of any function. If someone stopped you on the street, pointed a gun at you and said

"Here, I have this function: $$h(a) = \frac{2a+1}{3},$$ I need the formula for $h^{-1}$. Give it to me or I'll shoot you!"

then you don't need to know where that function came from, all you need to do is figure out the inverse: $$\begin{align*} b &= \frac{2a+1}{3}\\ 3b &= 2a+1\\ 3b-1 &= 2a\\ &\vdots \end{align*}$$ etc. When you are done and have a formula for $h^{-1}(a) = (g\circ f)^{-1}(a)$, you can compare it to the formula you found for $f^{-1}\circ g^{-1}$ and verify that you got the same function.

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You find the inverse of $g\circ f$ by using the fact that $(g\circ f)^{-1} = f^{-1} \circ g^{-1}$.

In other words, what gets done last gets undone first.

$f$ multiplies by 2 and then adds 1.

$g$ divides by 3.

Dividing by 3 is done last, so it's undone first.

The inverse first multiplies by 3, then undoes $f$.

Later note: Per the comment, to verify that $(g\circ f)^{-1} = f^{-1} \circ g^{-1}$:

Instead of confusingly writing $a = g(b)$, write $c=g(b)$. Then $c=b/3$, so $b=3c$, so $$g^{-1}(c) = 3c.$$ And $$f^{-1}(b) = \frac{b-1}{2}.$$

So $$ b = 3c\qquad\text{and}\qquad a = \frac{b-1}{2}. $$ Put $3c$ where $b$ is and get $$ a=\frac{3c-1}{2}. $$ You want to show that that's the same as what you'd get by finding $g(f(a))$ directly and then inverting.

So $c = g(f(a)) = \dfrac{f(a)}{3} = \dfrac{2a+1}{3}$.

So take $c = \dfrac{2a+1}{3}$ and solve it for $a$:

$$ \begin{align} 3c & = 2a+1 \\ 3c - 1 & = 2a \\ \\ \frac{3c-1}{2} & = a. \end{align} $$

FINALLY, observe that you got the same thing both ways.

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