In view of the coronavirus pandemic, we are making LIVE CLASSES and VIDEO CLASSES completely FREE to prevent interruption in studies
Maths > Relations and Functions > Algebra of Real Functions
Relations and Functions

Algebra of Real Functions

Assume a case where Seema, whose monthly income is Rs. 15,000, spends Rs. 10,000. What will be her saving? Simple! Rs. 5,000. Saving = Income – Expenditure. Here, we see that the input and the output are the real numbers. We can say that real number input gives a real number output. Here, we will learn Real-valued functions and algebra of real functions. The above case is a representation of real mathematical functions and a case of subtraction in the algebra of real functions.

Suggested Videos

Play
Play
Play
Arrow
Arrow
ArrowArrow
Cartesian Product
Introduction to Relations
Functions
Slider

 

Real-valued Mathematical Functions

In mathematics, a real-valued function is a function whose values are real numbers. It is a function that maps a real number to each member of its domain. Also, we can say that a real-valued function is a function whose outputs are real numbers i.e., f: R(stands for Real).

mathematical functions

 Download Relations Cheat Sheet PDF by clicking on Download button below

relation cheat sheet

relation cheat sheet

 

Algebra of Real Functions

In this section, we will get to know about addition, subtraction, multiplication, and division of real mathematical functions with another.

Addition of Two Real Functions

Let f and g be two real valued functions such that f: X→and g: X→where X ⊂ R. The addition of these two functions (f + g) : X→R  is defined by:

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

Subtraction of One Real Function from the Other

Let f: X→and g: X→R be two real functions where X ⊂ R. The subtraction of these two functions (f – g): X→R  is defined by:

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

Multiplication by a Scalar

Let f: X→be a real-valued function and γ be any scalar (real number). Then the product of a real function by a scalar γf: X→is given by:

(γf) (x) = γ f(x), for all x ∈ X.

Multiplication of Two Real Functions

The product of two real functions say, f and g such that f: X→R and g: X→R, is given by

(fg) (x) = f(x) g(x), for all x ∈ X.

Division of Two Real Functions

Let f and g be two real-valued functions such that f: X→and g: X→where X ⊂ R. The quotient of these two functions (f  ⁄ g): X→R  is defined by:

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

Note: It is also called pointwise multiplication.

Solved Example for You

Question 1: Let f(x) = xand g(x) = 3x + 1 and a scalar, γ= 6. Find

  1. (f + g) (x)
  2. (f – g) (x)
  3. (γf) (x)
  4. (γg) (x)
  5. (fg) (x)
  6. (f / g) (x)

Answer : We have,

  1. (f + g) (x) = f(x) + g(x) = x+ 3x + 1.
  2. (f – g) (x) = f(x) – g(x) = x– (3x + 1) = x– 3x – 1.
  3. (γf) (x) = γ f(x) = 6x
  4. (γg) (x) = γ g(x) = 6 (3x + 1) = 18x + 6.
  5. (fg) (x) = f(x) g(x) = x(3x +1) = 3x4 + x3.
  6. (f / g) (x) = f(x) / g(x) = x/ (3x +1), provided x ≠ – 1/3.

Question 2: What is meant by functions in algebra?

Answer: A function refers to an equation that consists of only one answer for y for every x. A function assigns only one output to each input that is associated with a specified type. It is common that a function is named as g(x) or f(x) but not y.

Question 3: Explain what makes a function a function?

Answer: A relation from a set X to a set Y is known as a function in case each element of X has a relation to exactly one element in Y. For example, consider an element x in X, so there shall be only one element in Y that x can have a relation to.

Question 4: Is it possible for an equation to be a function?

Answer: An equation shall be considered a function only when for every x’s value there is only one corresponding value for y.

Question 5: When will a function be well defined?

Answer: A function will be well defined when it provides the same result when a change takes place in the representation of the input without the change taking place in the value of the input.

Share with friends

Customize your course in 30 seconds

Which class are you in?
5th
6th
7th
8th
9th
10th
11th
12th
Get ready for all-new Live Classes!
Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes.
tutor
tutor
Ashhar Firdausi
IIT Roorkee
Biology
tutor
tutor
Dr. Nazma Shaik
VTU
Chemistry
tutor
tutor
Gaurav Tiwari
APJAKTU
Physics
Get Started

Leave a Reply

avatar
  Subscribe  
Notify of

Stuck with a

Question Mark?

Have a doubt at 3 am? Our experts are available 24x7. Connect with a tutor instantly and get your concepts cleared in less than 3 steps.
toppr Code

chance to win a

study tour
to ISRO

Download the App

Watch lectures, practise questions and take tests on the go.

Get Question Papers of Last 10 Years

Which class are you in?
No thanks.