Question

If the range of a function is a singleton set, then it is a
A) A constant function
B) An identity function
C) A bijective function
D) An one-one function

Answer 1

Hint: In this question, we have to choose the correct option for the particular required.
The range of a function is the set of all output values of a function that is the co-domain of the function or the image of the function.
We have to find out the type of the function for a singleton co-domain.

Complete step-by-step answer:
It is given that the range of a function is a singleton set.
Now, a singleton set is defined as a set that contains only one element.
We need to find the type of the function when the range of a function is a singleton set.
In mathematics, a constant function is a function whose output value is the same for every input value. That means the co-domain contains only one value.
In mathematics, an identity function is a function that always returns the same value that was used as its argument.
$f(x)=x$ for all x belongs to the domain. Thus the co-domain can contain more than one element.
Similarly one-one and bijective function can also contain more than one element.
Thus the function is a constant function.

The option (A) is the correct option.

Note: Constant function:
In mathematics, a constant function is a function whose output value is the same for every input value.
\[f(x) = c,\] for all \[x\] belongs to the domain. \[c\] is a constant.
Identity function:
In mathematics, an identity function is a function that always returns the same value that was used as its argument.
\[f(x) = x,\]for all \[x\] belongs to the domain.
One-one function:
A function \[f:X \to Y\] is said to be one to one (or injective function), if the images of distinct elements of \[X\] under \[f\] are distinct.
That is for every \[{x_1},{x_2} \in X,f\left( {{x_1}} \right) = f\left( {{x_2}} \right)\]implies \[{x_1} = {x_2}\].
Onto or surjective function:
In mathematics, a function \[f\] from a set \[X\] to a set \[Y\] is surjective, if for every element \[y\] in the codomain \[Y\] of \[f\], there is at least one element \[x\] in the domain \[X\] of \[f\] such that \[f\left( x \right) = y\]. It is not required that \[x\] be unique; the function \[f\] may map one or more elements of \[X\] to the same element of \[Y\].
A function that is one-one and onto is called bijective function.
That is a bijective function is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.