Question regarding inverse functions

[michellemich]

f(x) where x belongs to all real numbers
inverse: f-1(x), where x belongs to all real numbers

True or False:
The inverse of f(x+3) is f-1(x+3)

My ideas:
I think that it is false given that when you usually find the inverse of a function, you switch the x and y variables and solve for y again meaning that the inverse couldn't stay the same.
I figured since the domain and range of f(x) belong to all real numbers, possibly f(x) = x and then inputting f(x+3) = x+ 3
then y = x+3
then y = x - 3 but I am not really sure if that's right :s

 

[LCKurtz]

f(x) where x belongs to all real numbers
inverse: f-1(x), where x belongs to all real numbers

True or False:
The inverse of f(x+3) is f-1(x+3)

My ideas:
I think that it is false given that when you usually find the inverse of a function, you switch the x and y variables and solve for y again meaning that the inverse couldn't stay the same.
I figured since the domain and range of f(x) belong to all real numbers, possibly f(x) = x and then inputting f(x+3) = x+ 3
then y = x+3
then y = x - 3 but I am not really sure if that's right :s

You are given that $f$ has an inverse $f^{-1}$. What happens when you solve the equation $y=f(x+3)$ for $x$?

 

[BrettJimison]

Good Day michellemich!

If you are not sure of your answer, try some composition: let your original function be f(x)and your questionable inverse function be g(x)

Evaluate (f of g) and (g of f). If they undo each other, they are inverses.

 

[HallsofIvy]

If you want to know if this is true for all invertible functions, it is simple enough to find a counterexample.

If, say, f(x)= 2x+ 3, then $f(x)= 3x- 2$, then $f^{-1}(x)= (x+ 2)/3$. f(x+3)= 3(x+ 3)- 2= 3x+ 7. The inverse of that function is $(x- 7)/3$. Is that equal to $f^{-1}(x+ 3)= (x+3+ 2)/3= (x+ 5)/3$?