Question

The notation f:S→T denotes that f is a function, also called a map , defined on all of a set S and whose outputs lie in a set T . A function f:S→T is injective if for all x,y∈S , f(x)=f(y) implies that x=y . Alternatively: a function is injective if we can uniquely recover some input x based on an output f(x) . What functions are injective?

Answer 1

There are many. Two examples are

`f(x) = x, \\f(x) = x^3`

Explanation:

There are many examples. The simplest is

1 -

`f(x) = x`

It is trivial that

`\text{if \,\,\,\,} f(x) = f(y) \,\,\,\,\,\text{then} \,\,\,\,\, x=y`

2 -

`f(x) = x^3`

That function is injective as well.

`\text{if \,\,\,\,} x^3 = y^3 \,\,\,\,\,\text{then} \,\,\,\,\, x=y`

An example of a function that is NOT injective is

`f(x) = x^2`

Notice that

`f(-2) = (-2)^2 = 2^2 = 4`