Q. 145.0( 1 Vote )

# Mark the correct alternative in each of the following:

Which of the following functions from Z to itself are bijections?

A. f(x) = x^{3}

B. f(x) = x + 2

C. f(x) = 2x + 1

D. f(x) = x^{2} + x

Answer :

a. f(x) = x^{3}

⇒ 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) = x^{2} + 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.

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