Question

Show that even functions have no inverse, but odd functions may.

Answer 1

a function is even if f(-x) = f(x)

lets say a function is f(x) = x^2

this is an even function since f(-x) = f(x)

when finding inverse of this function

we get

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

+- sqrt (x) is not a function , it should either be + sqrt x or - sqrt x

hence, even functions have no inverse

whereas odd functions are functions when f(-x) = - f(x)

for example: f(x) = x^3

the inverse of this function is

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

which is a function

so, odd functions may have inverse