Injectivity:
Consider x, y ∈ R where f(x) = f(y)
We get
x5 = y5
It can be written as
x5 – y5 = 0
By multiplying and dividing by 2 on both sides
(x5/2)2 – (y5/2)2 = 0
We know that
(x5/2 + y5/2) (x5/2 – y5/2) = 0
So we get
x5/2 – y5/2 = 0
where x = y
Hence, f(x) = f(y) is x = y for all x, y ∈ R
f is injective.
Surjectivity:
Consider y as an arbitrary element of R
We know that
f(x) = y
It can be written as
x5 = y
So we get
x5 – y = 0
Odd degree equation has one real root
Thus, for every real value of y
x5 – y = 0 has real root α where
α 5 – y = 0
So α 5 = y
Thus, f(α) = y
For every y ∈ R there exists α ∈ R where f(α) = y
f is surjective
Thus, f: R → R is bijective.