Determine if each function from R R is injective surjective

Determine if each function from R -> R is injective, surjective, or bijective f(x) - -3x + 4 f(x) = :x^2 + 3 f(x) = x^3 f(x) = (x^2 + 1)(x^2 + 2) f(x) = tan(x)

Solution

Post one more question to get the remaining last two answers

The function is said to be injective if the function is one-one

The function is said to be surjective if the function is onto, if every element of the codomain is mapped to the least element of the domain

The function is said to be bijective if the function is both one-one and onto

a) f(x) = -3x+4

f(x1) = -3x1 + 4

f(x2) = -3x2 + 4

-3x1 + 4 = -3x2 + 4

x1 = x2

Hence the function is one-one

The function is onto since each and every element of the range R is covered

Therefore, the function is injective,surjective and bijective

b) f(x) = x^2 + 3

f(x1) = x1^2 + 3

f(x2) = x2^2 + 3

f(x1) = f(x2)

x1^2 + 3 = x2^2 + 3

x1 = x2 or (x1=-x2)

Hence the function is not one-one

The function is not onto, since f(x) is always greater than or equal to 3, since x^2 will be always positive

Hence the function is neither injective nor surjective

c) f(x) = x^3

f(x1) = x1^3

f(x2) = x2^3

x1^3 = x2^3

implies x1 = x2

Hence the function is one-one

Function is onto as well

Hence the function is both one-one and onto