Question

if an arbitrary function k and an arbitrary function l are 1-1 functions, use the definition to prove that k(l(x)) is 1-1.

Answer 1

Proof by contradiction: Suppose that k(l(x)) is not injective, so that there exist distinct x,y so that k(l(x)) = k(l(y)). Then because k is injective, l(x)=l(y). Then because l is injective, x=y, which contradicts our original condition. Therefore k(l(x)) is injective.

Answer 2

is that one minus one? or are you just abbreviating one-to-one?

Answer 3

define \[f(x)=k(l(x))\]show that if \(f(y)=f(x)\) then \(y=x\)