A function is bijective if for every value of x there exists a value of f(x), for every value of f(x) there exists a value of x and there is a one to one correspondence between x and f(x).
Here the function given is f(x) = x/(3x + 1) where x can take any real value.
For the value x = -1/3, f(x) = (-1/3)/(3*(-1/3) + 1) is of a form that is not defined. f(x) does not lie in R for x = -1/3 or there does not exist a valid value for f(x) if x = -1/3.
Therefore the function is not bijective.
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.