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:**

*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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