Question 7
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 + x^{2 }

Answer

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

.

∴ *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 x_{1 }= x_{2}

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.