Check whether f is a bijective function or not...Let f:R---->R be defined as f(x)=x/(3x+1) Check whether f is a bijective function or not?

For a function to be bijective, it has to be injective and
surjective at the same time.

We'll verify if the function
is injective.

We'll calculate the 1st
derivative:

f'(x) = [x'*(3x+1) -
x*(3x+1)']/(3x+1)^2

f'(x) = (3x + 1 -
3x)/(3x+1)^2

We'll eliminate like
terms:

f'(x) = 1/(3x+1)^2

We
notice that f'(x) is positive, therefore the function is strictly
increasing.

A strictly monotonic function is
injective.

We'll check if the functino is
surjective.

We'll evaluate the limit of f(x), if x
approaches to - infinite:

lim f(x) = lim x/(3x + 1) =
1/3

We notice that the function is not continuous towards -
infinite, therefore the function is not
surjective.

Since the function is not
surjective, therefore the function is not
bijective.