Mark the correct alternative in each of the following:
Which of the following functions from Z to itself are bijections?
a. f(x) = x3
⇒ For no value of x ϵ Z, f(x) = 2.
Hence, it is not bijection.
b. f(x) = x + 2
If f(x) = f(y)
⇒ x + 2 = y + 2
⇒ x = y
So, f is one-one.
Also, y = x + 2
⇒ x = y – 2 ϵ Z
So, f is onto.
Hence, this function is bijection.
c. f(x) = 2x + 1
If f(x) = f(y)
⇒ 2x + 1 = 2y + 1
⇒ x = y
So, f is one-one.
Also, y = 2x + 1
⇒ 2x = y – 1
So, f is into because x can never be odd for any value of y.
d. f(x) = x2 + x
For this function if we take x = 2,
f(x) = 4 + 2
⇒ f(x) = 6
For this function if we take x = -2,
f(x) = 4 - 2
⇒ f(x) = 2
So, in general for every negative x, f(x) will be always 0. There is no x ϵ R for which f(x) ϵ (-∞, 0).
It is not bijection.