Question

f is a linear function, f (x) = mx + c. a. Find f (x) if it is equal to its inverse: f ‐ ¹ (x) = f (x). b. Find f (x), if f (f (x)) = f (x)

This problem has got two solutions.

Answer 1

a. If $f$ maps $x$ to $y,$ then $f^{-1}$ maps $y$ to $x.$This gives rise to the cancellation formulas:

$f^{-1}(f(x))=x,$ ​for every $x$ in the domain of$f(x),$

$f(f^{-1}(x))=x,$ ​for every $x$ in the domain of$f^{-1}(x).$

Given $f^{-1}(f(x))=f(x).$ Then $f(x)=x.$

b. Given $f(f(x))=f(x).$ Then

$$m(mx+b)+b=mx+b$$

$$m^2x+(m+1)b=mx+b$$

$$m^2=m$$ $$m+1=1\ or\ b=0$$

Then $f(x)=b$ or $f(x)=x.$