Q. 395.0( 1 Vote )

# Which of the foll

A function is bijective iff it is one-one and onto.

Option A. f (x) = x3

Let f(x1) = f(x2)

x13= x23

x1= x2

f is one one

Let f(x) = y, y Z

y = x3

x = y1/3 but y1/3 Z

f is not onto

Thus, f is not bijective.

Option B. f (x) = x + 2

Let f(x1) = f(x2)

x1+2 = x2+2

x1= x2

f is one one

Let f(x) = y, y Z

y = x + 2

x = y – 2

for each y Z there exists x Z (domain) such that f(x) = y.

f is onto

Thus, f is bijective.

Option C. f (x) = 2x + 1

Let f(x1) = f(x2)

2x1+1 = 2x2+1

x1= x2

f is one one

Let f(x) = y, y Z

y = 2x + 1

y - 1 = 2x We observe that if we put y=0, then .

Thus, y = 0 Z does not have pre image in Z (domain)

f is not onto.

Thus, f is not bijective.

Option D. f (x) = x2 + 1

let f(x1) = f(x2)

x12 + 1 = x22 + 1

x12 = x22

x1 = ± x2

x1= x2 and x1= - x2

For e.g., f(-1) = |-1| = 1 and f(1) = |1| = 1

f is not one-one.

Since, f is not one one it cannot be bijective.

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