Question

Determine whether each of these functions is a bijection from R to R. a)f (x) = x^3

Answer 1

Solution.

$f(x)=x^3,f:R\to R$

Let x₁,x₂ ∈R and let us assume f(x₁)=f(x₂).

So, $x_1^3=x_2^3\implies x_1=x_2.$

Hence, we have f(x₁)=f(x₂)

​implies x₁=x_2.

So, f is one-one (injective).

Also we know

$$-\infty<x<\infty\implies\newline -\infty<x^3<\infty\implies\newline -\infty<f(x)<\infty\implies\newline$$

So, we clearly observe the Co-Domain is the same as the Range, so f(x) is surjective.

And hence f(x) is bijective.