This question is not particularly difficult, in terms of finding the answers. I want to make sure the "proof" is complete, in particular the part of "find all" questions where you need to show that these are the only functions that satisfy.
(High School Contest Math-Subjective) Find all linear functions $f: \mathfrak{R} \rightarrow \mathfrak{R}$ such that $f(x)=4f^{-1}(x)+27$
I can find the functions:
Substitute $f(x)=ax+b\implies f^{-1}(x)=\frac{x-b}{a}$ in the given condition:
$$ax+b=4(\frac{x-b}{a})+27=\frac{4x-4b+27a}{a}$$ Cross-multiply in the first and last equality: $$a^2x+ab=4x-4b+27a$$
Equating coefficients: $$x \text{ term: } a^2=4⇒a=±2$$ $$\text{Constant Term: }ab=27a-4b$$ $$\text{Case I}: a=2⇒2b=54-4b⇒6b=54⇒b=9⇒f_1 (x)=2x+9$$ $$\text{Case II}: a=-2⇒-2b=-54-4b⇒2b=-54⇒b=-27⇒ f_2 (x)=-2x-27$$
I need to demonstrate that these are the only functions that satisfy the given functional equation.
Is it sufficient to say that since I have chosen the most general form of a linear function, and substituted into the equation, I have found all the solutions? Or is anything further needed?