Find Expression for Function Evaluated at Inverse

Question

In my textbook, the following example question was phrased:

The answer is provided as:

I am fairly familiar with the inverse of basic functions, however the terminology 'evaluated at it's inverse', I have never seen before.

The text has also not at all touched upon the algebraic notion of f(f^-1(x)) or any occurrence of f, beyond the syntax f(x)= or f^-1(x)=

What's more concerning is the solution does not follow any of the rules of algebra that have been explained thus far.

The transition from f(x+1) = (x+2) - 2 is somewhat understandable given the example calls for f(x) = x-2. However the following steps arbitrarily remove x from the brackets as well as the values +2 and -2.

To my understanding:

(x+2)-2 != x

By way of the distributive property:

(x+2)-2 = -2x -4

Without any further explanation of the rules being applied here (and there isn't any) this question makes no sense.

Could someone fill in the blanks for me?

I have an assignment which also refers to syntax:

g(x) = f(x-z) + q

Here as well, the occurrence of f on the right side g(x) has never been discussed.

Answer 1

First: $(x+2)-2$ is not a multiplication but an addition, so we cannot apply distributivity, but we can apply associativity and we have:

$$ (x+2)-2=x+(2-2)=x+0=x $$

Second: $f^{-1}(f(x))=x$ is essentially the definition of inverse function.

Given $$f:X \to Y \qquad y=f(x)$$ if $f$ is one to one and $Y_1$ is its range, than the inverse function is defined as: $$ f^{-1}:Y_1 \to X \qquad f^{-1}(y)=x \quad \mbox{such that}\quad f(x)=y $$

I don't know if your book define the inverse function somewhere before the cited example, but, if yes, I suppose that it uses a similar definition.

So : $f^{-1}(y)=f^{-1}(f(x))=x$.