Let \(f(x) = \begin{cases} k(x) &\text{if }x>2, \\ 2+(x-2)^2&\text{if }x\leq2. \end{cases}\)Find the function k(x) such that f is its own inverse.

What do they mean by "k(x) such that f is its own inverse."

mathtoo
Aug 31, 2018

#1**+2 **

Look at Melody's graph here, Mathtoo : https://www.desmos.com/calculator/fpruev1hep

Note that that all the coordinates (a, b) on the graph have a corresponding point (b, a) on the graph

For instance....note that the point (0,6) is on the graph....and note also that the point (6, 0) is also on the graph

[ The inverse of a function just reverses the coordinates produced by that function ]

Does that make sense ???

CPhill
Aug 31, 2018

#4**+1 **

Another way to understand this question is this:

an "inverse" of a function f is a function f^{-1} such that for every number x, f(f^{-1}(x))=x=f^{-1}(f(x)). Some functions have an inverse functions and others don't.

So the meaning of "f is its own inverse" is that for every x, f(f(x))=x.

Guest Aug 31, 2018

#6

#10**0 **

try to substitute a number smaller than 2 for x in the equation f(f(x))=x and see what happens.

example: when we substitute 1 for x we get:

f(f(1))=f(3) (we know that f(1)=3 because we defined the function f for 2 and values that are smaller than 2 by f(x)=2+(x-2)^{2})

so:

f(f(1))=f(3)=1 (we know that f(f(1))=1 because f is it's own inverse, meaning that f(f(x))=x)

but we know that for values of x that are larger than 2, f(x)=k(x). 3 is larger than 2, so now we know that 1=f(3)=k(3)

so k(3)=1. Try to use that way to find k(x) for every value of x that is larger than 2.

nvm i see you got it

Guest Aug 31, 2018

edited by
Guest
Aug 31, 2018

edited by Guest Aug 31, 2018

edited by Guest Aug 31, 2018