what do you expect f(f^-1(x)) to equal?

Question

Support claim with examples and explain in words why this makes sense?

Answer 1

f(f^-1(x)) = x

For a specific example:

f(x) = y = 3x - 1

f^-1(x) is the inverse function

switch x and y

x=3y - 1

solve for y

y = (x+1)/3

f^-1(x) = (x+1)/3

f(f^-1(x)) = f((x+1)/3) = 3((x+1)/3) - 1 = x+1-1 = x

f(f^-1(x)) = f^-1(f(x)) = x

an inverse function is similar to an inverse arithmetic operation

x +3 -3 = x

x+3

x+3-3 = x

if you add and subtract the same number you go back to where you started.

f(f^-1(x)) is performing the reverse operation

3x/3 = x

multiply x by 3, then divide by 3, you go back to where you started: x

another specific example

f(x) = x +3

f^-1(x) = x-3

f(f^-1(x)) = f(x-3) = (x-3)+3 = x

or

f(x) = 3x

f^-1(x) = x/3

f(f^-1)(x) = f(x/3) = 3(x/3) = x