Real-valued and piece functions: The concept of function is a fundamental technique in almost all branches of Mathematics. In fact, they are the primary tool to describe the real world in mathematical notions. As a matter of fact, functions are some special kind of relations. There are so many types of the functions, such as real function, real-valued function, piece function, etc.

In everyday life, many quantities depend on one or more changing variables. For example, plant growth depends on sunlight and rainfall, speed depends on distance travelled and time taken, Voltage depends on current and resistance, and test marks depend on attitude, listening to lectures, and doing tutorials. Informally a function is a rule that relates to how one quantity depends on other quantities.

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Real Variable Function

The real variable function can be easily defined with the help of the concept of mapping. Let \({\mathbf{A}} \subseteq \mathbb{R}\) and \(B\) be any two non-empty sets. “A real variable function from \({\mathbf{A}} \subseteq \mathbb{R}\) and \(B\) is a rule or correspondence that assigns to each element of set \({\mathbf{A}} \subseteq \mathbb{R}\), one and only one element of set \(B\).

Let the correspondence be \(f\). Then, mathematically, we write \(f: {\mathbf{A}} \subseteq \mathbb{R} \to B\) where \(y = f(x), x ∈ {\mathbf{A}} \subseteq \mathbb{R} \to B\) and \(y∈B\). Here \(y\) is the image of \(x\) under \(f\) (or \(x\) is the pre-image of \(y\)). Set \({\mathbf{A}} \subseteq \mathbb{R}\) is called the domain of real variable function \(f\), and set \(B\) is called the co-domain of real variable function \(f\).

The set of images of different elements of set \({\mathbf{A}} \subseteq \mathbb{R}\) is called the range of real variable function \(f\). It is obvious that range will be a subset of co-domain as we may have a few elements in co-domain which are not the images of any element of set \({\mathbf{A}} \subseteq \mathbb{R}\)

Example: The function defined by \(f(x) = x + i\) is a real variable function.

Real-Valued Function

The real-valued function can be easily defined with the help of the concept of mapping. Let \(A\) and \({\mathbf{B}} \subseteq \mathbb{R}\) be any two non-empty sets. “A real-valued function from \(A\) and \({\mathbf{B}} \subseteq \mathbb{R}\) is a rule or correspondence that assigns to each element of set \(A\), one and only one element of set \({\mathbf{B}} \subseteq \mathbb{R}\). Let the correspondence be \(f\).

Then, mathematically, we write \(f: A \to {\mathbf{B}} \subseteq \mathbb{R}\) where \(y = f(x)\), \(x \in A\) and \(x \in {\mathbf{B}} \subseteq \mathbb{R}\). Here \(y\) is the image of \(x\) under \(f\) (or \(x\) is the pre-image of \(y\)). Set \(A\) is called the domain of real-valued function \(f\) and set \({\mathbf{B}} \subseteq \mathbb{R}\) is the co-domain of real-valued function \(f\).

The set of images of different elements of set \(A\) is called the range of real-valued function \(f\). It is obvious that range will be a subset of co-domain as we may have a few elements in co-domain which are not the images of any element of set \(A\). Real valued functions are classified as algebraic, trigonometric, logarithmic, etc.

Example: The function defined by \(f\left( {x + i} \right) = \sqrt {{x^2} + 1} \) is a real-valued function.

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Real Function

The real function can be easily defined with the help of the concept of mapping. Let \({\mathbf{A}} \subseteq \mathbb{R}\) and \({\mathbf{B}} \subseteq \mathbb{R}\) be any two non-empty sets. “A real function from \({\mathbf{A}} \subseteq \mathbb{R}\) and \({\mathbf{B}} \subseteq \mathbb{R}\) is a rule or correspondence that assigns to each element of set \({\mathbf{A}} \subseteq \mathbb{R}\) one and only one element of set \({\mathbf{B}} \subseteq \mathbb{R}\).

Let the correspondence be \(f\). Then, mathematically, we write \(f: {\mathbf{A}} \subseteq \mathbb{R} \to {\mathbf{B}} \subseteq \mathbb{R}\) where \(y = f(x)\), \(x \in {\mathbf{A}} \subseteq \mathbb{R}\) and \(y \in {\mathbf{B}} \subseteq \mathbb{R}\).

Here \(y\) is the image of \(x\) under \(f\) (or \(x\) is the pre-image of \(y\)). Set \({\mathbf{A}} \subseteq \mathbb{R}\) is called the domain of real function \(f\) and set \({\mathbf{B}} \subseteq \mathbb{R}\) is called the co-domain of real function \(f\). The set of images of different elements of set \({\mathbf{A}} \subseteq \mathbb{R}\) is called the range of function \(f\). It is obvious that range will be a subset of co-domain as we may have a few elements in co-domain which are not the images of any element of set \({\mathbf{A}} \subseteq \mathbb{R}\).

Example: The function defined by \(f(x) = x + 1\) is a real function

Difference between Real Function, Real-Valued Function and Real Variable Function

Following are the differences between Real Function, Real-Valued Function and Real Variable Function in detail.

Real Function	Real-Valued Function	Real Variable Function
If \(f: A \to B\) is a function such that \({\mathbf{A}} \subseteq \mathbb{R}\) and \({\mathbf{B}} \subseteq \mathbb{R}\), then \(f\) is called a real function.	If \(f: A \to B\) is a function such that \({\mathbf{B}} \subseteq \mathbb{R}\), then \(f\) is called a real-valued function.	If \(f: A \to B\) is a function such that \({\mathbf{A}} \subseteq \mathbb{R}\), then \(f\) is called a real variable function.
Domain and range should be the subsets of real numbers.	Range should be the subset of real numbers.	Domain should be the subset of real numbers.

Piecewise Functions

The piecewise-defined function (also called a piecewise function) is a function whose definition changes depending on the value of the independent variable. Piecewise or non-uniformly defined functions are the functions whose domain is divided into different parts. The function has different analytical formulas in the different parts of its domain.

We can say that a piecewise-defined function is composed of branches of two or more functions. Word piecewise is also used to describe the property of a piecewise-defined function which holds true for each piece but may not hold true for the whole domain of the function. In other words, we can say that a piecewise function is a function \(f(x)\) which has different definitions in different intervals of the domain of the function. The graph of a piecewise function has different pieces corresponding to each of its definitions.

The function shown in the graph is made up of three pieces.
To understand more about piecewise function, let us take a piecewise function

\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {{x^2} + 2,\;\;\;0 < x \leqslant 2} \\ {4,\;\;\;\;\;\;\;\;\;\;\;\;2 < x < 5} \\ {x,\;\;\;\;\;\;\;\;\;\;\;5 \leqslant x < 7} \end{array}} \right.\)

We should read this piecewise function as
\(f(x)\) is equal to \(x^2 + 2\) when \(x\) is greater than \(0\) and less than or equal to \(2\).
\(f(x)\) is equal to \(4\) when \(x\) is greater than \(2\) and less than \(5\).
\(f(x)\) is equal to \(x\) when \(x\) is greater than or equal to \(5\) and less than \(7\).
To draw the graph of the functions, follow the following steps:

Step 1: Write the intervals that are shown in the definition of the function along with their definitions,
Step 2: Draw the graph of the corresponding definition of the function in their corresponding intervals.

Graph of the function \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {{x^2} + 2,\;\;\;0 < x \leqslant 2} \\ {4,\;\;\;\;\;\;\;\;\;\;2 < x < 5} \\ {x,\;\;\;\;\;\;\;\;\;\;5 \leqslant x < 7} \end{array}} \right.\)

Domain and Range of a Piecewise Function

The domain of a piecewise-defined function is the union of its subdomains. The range of a piecewise-defined function is the union of the ranges of each subfunction over its subdomain.

Solved Examples – Real-Valued and Piece Functions

Below are a few solved examples that can help in getting a better idea.

Q.1. Determine the domain of a piecewise function
\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {2;\;\;\;\;\;\;\;\;\;x < 0} \\ { - 3;\;\; \;\;\;\;x > 0} \end{array}} \right.\)
Sol. Graph for the given function \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {2;\;\;\;\;\;\;\;\;\;x < 0} \\ { - 3;\;\; \;\;\;\;x > 0} \end{array}} \right.\)

The domain of the function is the set of values of input for which the function is defined, and the range of a function is the set of all possible outputs of the function, given its domain. The domain of \(f(x)\) by considering the subdomains of the function, \(x < 0\) and \(x > 0\), which both do not include \(0\). The union of these subdomains is the domain of the function, \(\mathbb{R} – \{ 0 \}\).

Q.2. Determine the range of a piecewise function
\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {x;\;\;\;\;\;\;\;\;\;\;\;\;x < 0} \\ { - 2x;\;\;\;\;\;\;\;\;\;x > 0} \end{array}} \right.\)
Sol. Graph for the given function \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {x;\;\;\;\;\;\;\;\;\;\;\;\;x < 0} \\ { - 2x;\;\;\;\;\;\;\;\;\;x > 0} \end{array}} \right.\) is

The domain of \(f(x)\) by considering the subdomains of the function, \(x < 0\) and \(x > 0\), which both do not include \(0\). The union of these subdomains is the domain of the function, \(\mathbb{R} – \{0\}\).
Range of a function is the set of all possible outputs of the function. So, the range of this function is \(\left( { – \infty ,\;0} \right)\).

Q.3. Determine whether the domain and range of the function \(f\left( {x + i} \right) = \sqrt {{x^2} + 1} \) are the subsets of the real number.
Sol. Let the function \(f: A \to B\) be defined as \(f(x + i) = \sqrt {{x^2} + 1} \).
Here, set \(A\) is called the domain of function \(f\) and set \(B\) is called the co-domain of function \(f\). The set of images of different elements of set \(A\) is called the range of function \(f\). It is obvious that range will be a subset of co-domain as we may have a few elements in co-domain which are not the images of any element of set \(A\). For this function, for every non-real element of the domain, there exists an image that belongs to the real numbers.
Hence, range is the set of real numbers, but the domain is not a real subset of real numbers.

Q.4. Determine whether the domain and range of the function \(f(x) = 3x + i\) are the subsets of the real number.
Sol. Let the function \(f: A \to B\) be defined as \(f(x) = 3x + i\).
Here, set \(A\) is called the domain of function \(f\) and set \(B\) is called the co-domain of function \(f\). The set of images of different elements of set \(A\) is called the range of real variable function \(f\). It is obvious that range will be a subset of co-domain as we may have a few elements in co-domain which are not the images of any element of set \(A\). For this function, for every real number, there exists an image that belongs to the complex numbers.
Hence, the domain of \(f(x) = 3x + i\) is a set of real numbers, and the range is not a subset of real numbers. This is a real variable function.

Q.5. Find the domain and range of the function \(f\left( x \right) = \sqrt {{x^2} – 5x + 6} \)
Sol. \(f\left( x \right) = \sqrt {{x^2} – 5x + 6} \); for domain \({x^2} – 5x + 6 \geqslant 0\)
\( \Rightarrow \left( {x – 2} \right)\left( {x – 3} \right) \geqslant 0\)
\( \Rightarrow x \leqslant 2\) or \(x \geqslant 3\) (\(\therefore\) either both factors are positive or both non-negative)
\(\therefore \,{D_f} = \left( { – \infty ,\;2} \right] \cup \left[ {3,\;\infty } \right)\)
Also, \(y = f\left( x \right) = \sqrt {{x^2} – 5x + 6} = \sqrt {\left( {x – 2} \right)\left( {x – 3} \right)} \); for \(x \geqslant 3\); \(\left( {x – 2} \right)\left( {x – 3} \right) \in \left[ {0,\;\infty } \right)\); also for \(x \leqslant 2\); \(\left( {x – 2} \right)\left( {x – 3} \right) \in \left[ {0,\;\infty } \right)\)
\( \Rightarrow f\left( x \right) = \sqrt {{x^2} – 5x + 6} \in \left[ {0,\;\infty } \right)\)
\(\therefore \,{R_f} = \left[ {0,\;\infty } \right)\)
Hence, \({D_f} = \left( { – \infty ,\;2} \right] \cup \left[ {3,\;\infty } \right)\) and \({R_f} = \left[ {0,\;\infty } \right)\).

Summary

If \(f: A \to B\) is a function such that \({\mathbf{A}} \subseteq \mathbb{R}\) and \({\mathbf{B}} \subseteq \mathbb{R}\), then \(f\) is called a real function. If \(f: A \to B\) is a function such that \({\mathbf{B}} \subseteq \mathbb{R}\), then \(f\) is called a real-valued function.

If \(f: A \to B\) is a function such that \({\mathbf{A}} \subseteq \mathbb{R}\), then  is called a real variable function. Piecewise or non-uniformly defined functions are the functions whose domain is divided into different parts such that the function has different analytical formulae in the different parts of its domain. The domain of a piecewise-defined function is the union of its subdomains. The range of a piecewise-defined function is the union of the ranges of each subfunction over its subdomain.

FAQs on Real-Valued and Piece Functions

Students might be having many questions regarding the Real-Valued and Piece Functions. Here are a few commonly asked questions and answers.

Q.1. What is the Real Function of Real-Valued Mathematical Functions?
Ans: If \(f: A \to B\)is a function such that \({\mathbf{A}} \subseteq \mathbb{R}\) and \({\mathbf{B}} \subseteq \mathbb{R}\), then \(f\) is called a real function, and If \(f: A \to B\) is a function such that \({\mathbf{B}} \subseteq \mathbb{R}\), then \(f\) is called real-valued function.

Q.2. How are the real-valued functions classified?
Ans: Real valued functions are classified based on the type of mathematical equations. Some are algebraic, and some others are trigonometric, while some real-valued functions are logarithmic.

Q.3. What is the range of the real function?
Ans: The range of the real function is a subset of real numbers.

Q.4. How to read piecewise functions?
Ans: Let the piecewise function is \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {{x^2} + 2,\;\;\;\;\;\;\;\;0 < x \leqslant 2} \\ {4,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2 < x < 5} \\ {x,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;5 \leqslant x < 7} \end{array}} \right.\)

Then we read this piecewise function as
\(f(x)\) is equal to \(x^2 + 2\) when \(x\) is greater than \(0\) and less than or equal to \(2\).
\(f(x)\) is equal to \(4\) when \(x\) is greater than \(2\) and less than \(5\).
\(f(x)\) is equal to \(x\) when \(x\) is greater than or equal to \(5\) and less than \(7\).

Q.5. How to solve piecewise functions?
Ans: To solve the piecewise function, follow the following steps:
Step 1: Write the intervals shown in the definition of the function and their definitions.
Step 2: Solve the corresponding definition of the function in their corresponding intervals.

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