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

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=x2.

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.