Consider the sets A and Z related to each other as shown below: Clearly, every element in set A is related to exactly one element in the set Z. So, the given relation is a function.

The element 2 in the input set is assigned the element C from the output set.

$\boldsymbol{f(2) = C}$

Here, C is called the image of 2 under $f$ and 2 is called the preimage of C under $f$. The function $f$ from an input set A to an output set B is donated by

$\boldsymbol{f: A \rightarrow B}$

Here, A is the domain of the function and B is the codomain of the function.

A function whose range is a set of real numbers is called a real-valued function.

Example 1: Consider the sets D and Y related to each other as shown below. Can we consider this relation as a real-valued function? Solution:

Clearly, every element in the set D is related to exactly one element in the set Y. So, the given relation is a function.

The range of this function consists of the elements 2, 3, 5, and 7.

Range = {2, 3, 5, 7}

Hence, the range of this function is a set of real numbers. So, this function is a real-valued function.

Example 2: Consider the sets D and Y related to each other as shown below. Can we consider this relation as a real-valued function? Solution:

Clearly, every element in the set D is related to exactly one element in the set Y. So, the given relation is a function.

The range of this function consists of the elements 5, 6, 7, and 9.

Range = {5, 6, 7, 9}

Hence, the range of this function is a set of real numbers and therefore, this function is a real-valued function.

Example 3: Consider the sets D and Y related to each other as shown below. Can we consider this relation as a real-valued function? Solution:

Clearly, every element in the set D is related to exactly one element in the set Y. So, the given relation is a function.

The range of this function consists of the elements K, 6, 8, and 9.

Range = {K, 6, 8, 9}

Since ‘K’ is not a real number, the range is not the set of real numbers and therefore, this function is not a real-valued function.

Example 4: Consider the sets D and Y related to each other as shown below. Can we consider this relation as a real-valued function? Solution:

Clearly, every element in the set D is related to exactly one element in the set Y. So, the given relation is a function.

The range of this function consists of the elements $\sqrt{2}$, 3, 8, and 9.

Range = {$\sqrt{2}$, 3, 8, and 9}

Since the range of this function is a set of real numbers, therefore, this function is a real-valued function.

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Real-Valued Functions