Question

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i) f: R → R defined by f(x) = 3 − 4x

(ii) f: R → R defined by f(x) = 1 + x²

Answer 1

(i) f: R → R is defined as f(x) = 3 − 4x.

.

∴ f is one-one.

For any real number (y) in R, there existsin R such that

∴f is onto.

Hence, f is bijective.

(ii) f: R → R is defined as

.

.

∴does not imply that

For instance,

∴ f is not one-one.

Consider an element −2 in co-domain R.

It is seen thatis positive for all x ∈ R.

Thus, there does not exist any x in domain R such that f(x) = −2.

∴ f is not onto.

Hence, f is neither one-one nor onto.