Question

In each of the following cases state whether the function is bijective or not. Justify your answer. 

(i) f: R → R defined by f (x) = 2x + 1 

(ii) f: R → R defined by f(x) = 3 – 4x²

Answer 1

(i) f(x) = 2x + 1 

f(0) = 2(0) + 1 = 0 + 1 = 1 

f(1) = 2(1) + 1 = 2 + 1 = 3 

f(2) = 2(2) + 1 = 4 + 1 = 5

f(3) = 2(3) + 1 = 6 + 1 = 7

Different elements has different images 

∴ It is an one-one function. 

It is also an onto function. The function is one-one and onto function. 

∴ It is a bijective function.

(ii) f(x) = 3 – 4x² 

f(1) = 3 – 4(1)² 

= 3 – 4 = -1 

f(2) = 3 – 4(2)² = 3 – 16 = – 13 

f(3) = 3 – 4(3)² = 3 – 36 = – 33 

f(4) = 3 – 4(4)² = 3 – 64 = – 61 

It is not a bijective function. The positive numbers “R” do not have negative pre – image in X in R.