Real Valued Functions: Types, Operations & Examples

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A real valued function is the function that assigns real numbers to all the members of its domain. In simple words, real numbers are given as actual values to all the members of the domain.

Real valued functions of a real variable are the main objective of the calculus functions.
It is also known as real functions.

The real-valued function is real numbers assigned to members of a domain that includes all the real numbers, including the integers. It describes the topological space and complete metric space that find the use of these functions.

It is an important concept that is used in Sets and Functions.
The mapping of the subset of a set R of all the real numbers into R is the real-valued function of the real numbers.

Table of Content

What is a Real Valued Function?
What is a Subset?
What is a Proper Set?
Domain and Range of Real Valued Function
Types of Real Valued Function
Properties of Real Functions
Operations on Real Functions
Things to Remember
Sample Questions

Key Terms: Real Valued Functions, Real Functions, Subset, Proper Set, Domain, Range, One-to-one function, Many-to-one function, Into function, Onto function, Polynomial function, Linear function, Quadratic function, Identical function, Constant function

What is a Real Valued Function?

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Real-Valued Function, also known as real functions, is a type of function whose range lies within the real numbers, which include non-root numbers and non-complex numbers.

It involves the mapping of functions whose domain and codomain are subsets of the set of real numbers.
The concept is used for the formation of topological space and a complete metric space.

A function that has either R or one of its subsets as its range is called a real-valued function. If the domain of a set is also either R or a subset of R then they are real functions.

Real valued functions are used for assigning values to arguments.
The range is a set of values obtained by the function.

Example of Real Valued Function

Example: A function f(n) = 2n, in which n= 0,1,2,3….. In this example f(n) is the real function in which the mapping of set R’ of all integers into R’ is assigned. Real numbers are given as real values to the set R.

What is a Subset?

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Subset is the set whose elements are members of some other set. In the field of set theory, a subset is denoted by the symbol ⊆ and read as ‘is a subset of’.

It can contain all the elements that are found in the set.
Proper subsets and improper subsets are two types of subsets.
An empty set is consisdered subset of every given set.
Power set refers to the collection of all subsets.
A simpler version is ‘’ all the members of A are the members of C then A is the subset of C’’.

Example of What is a Subset?

Example: For example, if set A = {3, 4, 6}, then,

Number of subsets: {3}, {4}, {6}, {3,4}, {4,6}, {3,6}, {3,4,6} and Φ or {}.

What is a Proper Set?

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A proper set of domains involves two sets, A and B, in which the elements of B are equally present in set A, but set A is slightly different in that it contains at least one element more that is not present in set B.

In simpler terms, proper subset refers to a set that contains some, but not all, of the members of another set.
It is obtained by picking a subset of items from a larger set.
The set will remove at least one member from the larger set.
Mathematically, it can be represented as:

∀x (x ∈ A → x ∈ B) ∧ ∃y (y ∈ B ∧ y ∉ A)

Example of What is a Proper Set?

Example: If A = (a,b,c) and B = (b,c) Then B is the proper subset of A.

Domain and Range of Real Valued Function

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The domain and range of a real-valued function are determined on the basis of the given function to be defined in the real set. The domain of a function is said to be the set of every plausible input of the given function.

The range of a function is a set of all outputs of a function.
It is calculated by substituting the values of x to find the y-values.
Co-domain of a function is the set of all possible outputs of the function.
In the case of the exponential function, the domain is equal to all real lines, and the range is all positive real values.

Example of Domain and Range of Real Valued Function

Example: Find the domain and range of a function f(x) = 2x² – 5.

Ans: Given function: f(x) = 2x² – 5

Since domain of a function is the set of all input values for f, in which the function is real and defined.

The domain is the set of all real numbers.

Domain = [-∞, ∞]

Also, the range of a function is as follows:

Let y = 2x² – 5
2x² = y + 5
x² = (y + 5)/2
x = √[(y + 5)/2]
Square root function will be defined for non-negative values.
So, √[(y + 5)/2] ≥ 0
This is possible when y is greater than y ≥ -5.
Hence, the range of f(x) is [-5, ∞).

Types of Real Valued Functions

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Types of real valued functions are divided into two categories which are as follows:

Types of Functions involved in the Set

On the basis of types of functions involved in the set are as follows:

One-to-One Function

One-to-One Function is the type of function that occurs between two sets A and B. The mapping of each element of set A to the elements of set B.

Every element of two sets is mapped to another set. It is a one-to-one mapping.
One-to-one function is called an injective function.

Example of One-to-One Function

Example: Let A = {2, 4, 8, 10} and B = {w, x, y, z}. Which of the following sets of ordered pairs represent a one to one function?

{(2, w), (2, x), (2, y), (2,z)}
{(4,w), (2,x), (10,z), (8, y)}
{(4,w), (2,x), (8,x), (10, y)}

Ans: In the case of the second option there is a special element of A for every special element from B, therefore {(4,w), (2,x), (10,z), (8, y)} is representing an one to one function.

One-to-One Functions

Many-to-One Function

Many-to-One Function includes mapping of two or more elements of one function or set to only one element of another set or function. Consider ‘x’ a function whose two or more elements are mapped to one element of another function ‘y’.

It is the inverse form of one-to-one function

Example of Many-to-One Function

Example: Find the domain and range of the many one function f = {(1, a), (2, a), (3, a), (5, b), (6, c)}

Ans: The given function is f = {(1, a), (2, a), (3, a), (5, b), (6, c)}

Here we have:

Domain = (1, 2, 3)

Range = (a, b, c)

We observe that the elements 1, 2, 3 are connected to the same element 'a' in the range set. Hence function connecting the elements of the domain set to the element in the range set, makes a many one function.

Many to One Function

Many to one Functions

Into Function

Into function is mapping of elements from one set to every single image of another set. There exists at least one element left without pre-image in the previous set.

The mapping is into mapping and the function ‘f’ is said to be into function.

Example of Into Function

Example: Consider, A = {a, b, c}

B = {1, 2, 3, 4} and f: A → B such that

f = {(a, 1), (b, 2), (c, 3)}

In the function f, the range i.e., {1, 2, 3} ≠ co-domain of Y i.e., {1, 2, 3, 4}

Into Functions

Onto Function

In mathematics, onto function is termed as ‘’surjective’’ function. It involves every element of a set there exists one or more elements mapped to.

Consider two sets A and B and if for every element of ‘B’ there exists one or more element matching with set ‘A’.
The Function is called onto function or surjective function.

Example of Onto Function

Example: Let A = {1, 2}, D = {4, 5} and let g = {(1, 4), (2, 5)}. Show that the function g is an onto function from C into D.

Ans: Domain = set A = {1, 2}

We can see that the element from C,1 has an image 4, and both 2. Thus, the Range of the function is {4, 5} which is equal to D. As a result g: C →D is an onto function.

Onto Function

Other Types of Real Valued Functions

Some other types of real valued functions are as follows:

Polynomial function

Polynomial function is the function that involves only the positive integer exponents or the non-negative integer’s exponents of a real variable.

Polynomial function

Linear Function

The graph of the linear function is a straight line. The function carries one independent variable and one dependent variable. Assume the following function as Y = f(x) = a + bx

The notion ‘x ‘is the independent variable whereas the notion ‘y’ is considered to be a dependent variable.

linear function

Quadratic function

Quadratic function is a polynomial of degree two and can be represented in the form F(x)= ax²+bx+c. A, B and C are different numbers not equal to zero.

The quadratic function always has a curved parabola open or closed form.

Quadratic function

Identical function

A function is said to be identical if the domain and co-domain are same or identical. Identical functions are also known as equal functions as they are similar with domains and codomains.

Some of functions to be identical are represented as follows:
f (1) = g(1), f(-9) = g(-9) and so on…..

Constant function

The constant function is explained as the function whose output value remains the same as every input value. There is no change with output regardless of input.

Example of Constant Function

Example: Consider a function Y(X) = 5. It is the constant function as the value of ‘x’ is 4.

Properties of Real Functions

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The properties of real functions are as follows:

The numerator and denominator of a function must follow Hurwitz polynomials.
The degree of the required numerator of function must not be greater than the degree of the denominator by more than 1.
If the given function f(x) is a positive function, then its reciprocal is also a positive function.
The addition of two positive functions is also a positive function.
However, in the case of the subtraction function, the result will be either a positive or negative function.

Operations of Real Functions

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The operation of real functions are as follows:

Addition of Two Real Functions

Let f and g be two different real-valued functions, then f: X → R and g: X → R be any two real functions, where X ⊂ R. Then, the addition of two real functions are given by (f + g): X → R, which is as follows:

(f + g) (x) = f (x) + g (x), for all x ∈ X

Example of Addition of Two Real Functions

Example: If f(x) = 3x² and g(x) = 3x are two real functions, then find (f + g)(x).

Ans: Given,

f(x) = 3x²
g(x) = 3x
Now, (f + g)(x) = f(x) + g(x) = 3x² + 3x

Subtraction of a Real Function

Let f and g be two different real-valued functions, then f : X → R and g: X → R be any two real functions, where X ⊂ R.Then, the subtraction of two real functions are given by (f - g): X → R, which is as follows:

(f – g) (x) = f(x) –g (x), for all x ∈ X

Example of Subtraction of a Real Function

Example: If f(x) = x² + 5x + 9 and g(x) = 17x are two real functions, then find (f – g)(x).

Solution: Given,

f(x) = x² + 5x + 9
g(x) = 17x
(f – g)(x) = f(x) – g(x) = x² + 5x + 9 – (17x)
x² + 5x + 9 – 17x
x² + (5 – 17)x + (9)
x² – 12x + 9

Multiplication by a Scalar

Let f be a real-valued functions, then f: X → R and α be a scalar where scalar mean a real number. Then, the product α f is a function from X to R, i.e. (α f): X → R is given by

(α f) (x) = α f(x), x ∈ X

Example of Multiplication by a Scalar

Example: If f(x) = x² + 2x + 1, then find (α f)(x) such that α = 4.

Ans: Given,

f(x) = x² + 2x + 4

α = 4

(α f)(x) = α f(x) = 4(x² + 2x + 4) = 4x² + 8x + 16

Multiplication of Two Real Functions

Let f and g be two different real-valued functions, then f : X → R and g: X → R be any two real functions, where X ⊂ R.Then, the multiplication of two real functions are given by (fg): X → R, which is as follows:

(fg) (x) = f(x) g(x), for all x ∈ X

Example of Multiplication of Two Real Functions

Example: If f(x) = 5x and g(x) = x² are two real functions, then find (fg)(x).

Ans: Given,

f(x) = 5x
g(x) = x²
Now,
(fg) (x) = f(x) g(x)
(5x)(x²)
(5x)(x²)
5x³

Division of Two Real Functions

Let f and g be two different real-valued functions, then f : X → R and g: X → R be any two real functions, where X ⊂ R.Then, the division of two real functions are given by (f/g): X → R, which is as follows:

(f/g) (x) = f(x) / g(x), for all x ∈ X

Example of Division of Two Real Functions

Example: If f(x) = 5x³and g(x) = x² are two real functions, then find (fg)(x).

Ans: Given,

f(x) = 5x³
g(x) = x²
Now,
(f/g) (x) = f(x)/ g(x)
(5x³)/(x²)
5x

Things to Remember

The real valued function is a type of function where the range lies within the real numbers.
These functions are divided on the basis of sets and other functions.
Addition, subtraction, multiplication and division are four operations of real-valued functions.
It is an important concept of Sets and functions.
The function is based on the concept of calculus.

Sample Questions

Ques: Explain: (A) How are the real valued functions classified?
(B) What is the range of the real function? (2 marks)

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.

(B) Range of the function is the set of all values that ‘f’ takes. It is the elemental values mapped to the domain of a function.

Ques: Explain: (A) What is a preimage?
(B) Is the preimage and domain the same? (2 marks)

Ans. Preimage is the set of all values in the domain that is mapped to the elements in the range of the function.

(B) Yes, Preimage is the subset of domain. It includes all of the domain, some of the elements of the domain or sometimes none of the domain.

Ques. How do the domain and range are found? (2 marks)

Ans. Domains are the possible input values that are represented on the X-axis whereas the range is the set of all output values that is represented on the Y-axis in the graphical representation.

Ques. If f(x) = 3x – 4 and g(x) = x² – 2 are two real functions, then find (fg)(x)? (2 marks)

Ans. Given,

f(x) = 3x – 4
g(x) = x² – 2
Now,
(fg) (x) = g(x) g(x)
(3x – 4)(x² – 2)
(3x)(x²) + (3x)(-9) + (-4)(x²) + (-4)(-2)
3x³ – 27x – 4x² + 8

Ques. If f(x) = 3x² + 2x + 2, then find (α f)(x) such that α = 20? (2 marks)

Ans. Given,

f(x) = 3x² + 2x + 2

α = 20

(α f)(x) = α f(x) = 20(3x² + 2x + 2) = 60x² + 40x + 40

Ques. If f(x) = 10x²+8 and g(x) = 3x are two real functions, then find (f + g)(x)? (2 marks)

Ans: Given,

f(x) = 10x²+8
g(x) = 3x
Now, (f + g)(x) = f(x) + g(x) = 10x²+8 + 3x

Ques. Find the domain and range of a function f(x) = 2x² – 7? (3 marks)

Ans. Given function: f(x) = 2x² – 7

Since domain of a function is the set of all input values for f, in which the function is real and defined.

The domain is the set of all real numbers.

Domain = [-∞, ∞]

Also, the range of a function is as follows:

Let y = 2x² – 7
2x² = y + 7
x² = (y + 7)/2
x = √[(y + 7)/2]
Square root function will be defined for non-negative values.
So, √[(y + 7)/2] ≥ 0
This is possible when y is greater than y ≥ -7.
Hence, the range of f(x) is [-7, ∞).

Ques. If f(x) = x² + 5x and g(x) = 10x are two real functions, then find (f – g)(x)? (2 marks)

Ans. Given,

f(x) = x² + 5x
g(x) = 10x
(f – g)(x) = f(x) – g(x) = x² + 5x – (10x)
x² + 5x – 10x
x² + (5 – 10)x
x² – 5x

Ques. If f(x) = x² + 2x + 2, then find (α f)(x) such that α = 5? (2 marks)

Ans. Given,

f(x) = x² + 2x + 2

α = 5

(α f)(x) = α f(x) = 5(x² + 2x + 2) = 5x² + 10x + 10

Ques. If f(x) = 10x²+8 and g(x) = 10x +6 are two real functions, then find (f + g)(x)? (2 marks)

Ans: Given,

f(x) = 10x²+8
g(x) = 10x + 6
Now, (f + g)(x) = f(x) + g(x) = 10x²+8 + 6 +10x = 10x²+ 14 +10x

Ques. Find the domain and range of a function f(x) = 4x² – 7? (3 marks)

Ans. Given function: f(x) = 4x² – 7

Since domain of a function is the set of all input values for f, in which the function is real and defined.

The domain is the set of all real numbers.

Domain = [-∞, ∞]

Also, the range of a function is as follows:

Let y = 4x² – 7
4x² = y + 7
x² = (y + 7)/4
x = √[(y + 7)/4]
Square root function will be defined for non-negative values.
So, √[(y + 7)/4] ≥ 0
This is possible when y is greater than y ≥ -7.
Hence, the range of f(x) is [-7, ∞).

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Real Valued Functions: Types, Operations & Examples

What is a Real Valued Function?

Example of Real Valued Function

What is a Subset?

Example of What is a Subset?

What is a Proper Set?

Example of What is a Proper Set?

Domain and Range of Real Valued Function

Example of Domain and Range of Real Valued Function

Types of Real Valued Functions

Types of Functions involved in the Set

One-to-One Function

Example of One-to-One Function

Many-to-One Function

Example of Many-to-One Function

Into Function

Example of Into Function

Onto Function

Example of Onto Function

Other Types of Real Valued Functions

Polynomial function

Linear Function

Quadratic function

Identical function

Constant function

Example of Constant Function

Properties of Real Functions

Operations of Real Functions

Addition of Two Real Functions

Example of Addition of Two Real Functions

Subtraction of a Real Function

Example of Subtraction of a Real Function

Multiplication by a Scalar

Example of Multiplication by a Scalar

Multiplication of Two Real Functions

Example of Multiplication of Two Real Functions

Division of Two Real Functions

Example of Division of Two Real Functions

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1. If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that (i) \((A+B)'=A'+B' \)(ii) \((A-B)'=A'-B'\)

2. If (i) A=\(\begin{bmatrix} \cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha \end{bmatrix}\),then verify that A'A=I(ii) A= \(\begin{bmatrix} \sin\alpha & \cos\alpha\\ -\cos \alpha & \sin\alpha \end{bmatrix}\),then verify that A'A=I

3. Solve system of linear equations, using matrix method. x-y+2z=7 3x+4y-5z=-5 2x-y+3z=12

4. If \(\frac{d}{dx}f(x) = 4x^3-\frac{3}{x^4}\) such that \(f(2)=0\), then \(f(x)\) is

5. Find the inverse of each of the matrices,if it exists \(\begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix}\)

6. Find the vector and the cartesian equations of the lines that pass through the origin and(5,-2,3).

Similar Mathematics Concepts

Comments

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1.
If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that
(i) \((A+B)'=A'+B' \)
(ii) \((A-B)'=A'-B'\)

2.
If (i) A=\(\begin{bmatrix} \cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha \end{bmatrix}\),then verify that A'A=I
(ii) A= \(\begin{bmatrix} \sin\alpha & \cos\alpha\\ -\cos \alpha & \sin\alpha \end{bmatrix}\),then verify that A'A=I

3.
Solve system of linear equations, using matrix method.
x-y+2z=7
3x+4y-5z=-5
2x-y+3z=12

4.
If \(\frac{d}{dx}f(x) = 4x^3-\frac{3}{x^4}\) such that \(f(2)=0\), then \(f(x)\) is

5.
Find the inverse of each of the matrices,if it exists \(\begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix}\)

6.
Find the vector and the cartesian equations of the lines that pass through the origin and(5,-2,3).