Question

If $f:Z \to Z$ is such that $f(x) = 6x - 11$ then f is
(A) Injective but not surjective
(B) Surjective but not injective
(C) Bijective
(D) Neither injective nor surjective

Answer 1

Hint:Function – A relation f from a set A to set B is said to be a function, if every element of set A has one and only image in set B. In other words, a function f is a relation such that no two pairs in the relation have the first element.
Onto function – consider two sets, Set A and Set B, which consist of elements. If for every element of B, there is at least one or more than one element matching with A, then the function is said to be onto function or surjective function.
One-one function – It could be defined as each element of set A has a unique element on set B.


Complete step by step solution :
It is given that $f(x) = 6x - 11$.
We will differentiate a given function to get whether it is an increasing or decreasing function.
$F'(x) = 6(1) - 0$
$F'(x) = 6$
We know that,
If $f'(x) > 0$ at each point, the function is said to be increasing on Z.
So it is a one-one or injective function.
But f(x) cannot take all values in Z.
Hence, it is not onto or surjective.
Therefore, option A is a correct option.
Or, f is injective but not surjective.


Note : From the above solution we can say that the given function is injective but not surjective.
Do not get confused between the properties of injective and surjective functions
Let’s talk about bijective function.
Bijective function – A function is said to be bijective or bijection, if a function $f:A \to B$ satisfies both the injective (one-to-one function) and surjective function (onto function) properties.
And the above function is injective but not surjective.