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1800-102-2727We all are familiar with the fact that we can perform algebraic operations on numbers but now the question arises “ Can we perform algebraic operations on functions too? What are the conditions required to perform these operations?Can you visualize the resultant function?”
You will be able to answer all these questions after going through this article.
So in this article, we will learn about algebra of functions i.e., how to add two real functions, subtract a real function from another, multiply a real function by a scalar(real number), multiply two real functions and divide one real function by the other.
Table of contents
Let h: and be any two real functions, where . Then, we can define by , for all .
The domain of the function is
Example : Let and
Then
The domain of the function is . Now, the domain of both the functions is ,
= .
Subtraction of one real-valued function from another
Let and be any two real functions, where . Then, we can define by , for all .
The domain of the function (h-p)(x) is
Example:Let and
Then
Since D(h)= and D(p)= -{0},
D(h-p)= -{0} = -{0}.
Multiplication by a scalar
Let be a real valued function and be a scalar. Then, the product is a function from defined by , for all .
Note: The domain of function remains same as that of the original function in this case.
Example : Let and
Obviously domain of f is also .
Multiplication of two real functions
The product of two real functions and is a function defined by , for all . It is also called pointwise multiplication.
Note : If D(h) and D(p) are domains of the two functions h and p respectively then the domain of the functions (h+p), (h-p), hp will be
Example: Let f(x)=x+1 and
Then
(fg)(x)=
Since D(f)= and D(g)= -{0},
D(fg)= -{0} = -{0}.
Quotient of two real functions
Let and be any two real functions, where . Then, the quotient of h by p denoted by, is a function defined as , provided , .
Note: Domain of
Example : If and g(x)=x+2 then find and its domain.
Solution: We have,
Since D(f)= and D(g)= ,
as g(x)=0 at x=-2.
Note: The concept of the algebra of functions is used along with other concepts of mathematics like limits, continuity, differentiability, etc. where we need to combine two or more functions. Also, sometimes we break bigger functions into smaller ones with the help of the algebra of functions. Let us understand this with the help of some solved examples.
Example : Let f: be the function defined by &
g: be an arbitrary function. Let fg: be the product function defined by (fg)(x) = f(x)g(x). Then figure out which of the following statements are true ?
(A) fg is differentiable at x = 1 if g is continuous at x = 1
(B) g is continuous at x = 1 if fg is differentiable at x = 1
(C) fg is differentiable at x = 1 if g is differentiable at x = 1
(D) g is differentiable at x = 1 if fg is differentiable at x = 1
Solution :
and g:
……(i)
……(ii)
We know that sin x is a continuous function
Now,if the function g(x) is continuous at x=
Hence from (i) & (ii), h'(1+)=h'(1-)=(1+sin 1)g(1)
Therefore, h(x) is differentiable at x=1.
Also if g(x) is differentiable at is continuous at x=1
Therefore, h(x) will still be differentiable at x=1
Hence, options (A) and (C) are the correct answers.
Example : If and F(1)=2, then find F(101).
Solution :
Given,
On adding the above equations, we get
Example : If f (x) = cos (log x), then find the value of .
Solution :
f (x) = cos (log x)
Now let ,
y = 0.
Example : If . Find f(x).g(x).
Solution :
Drawing the graphs of f(x) & g(x)
Taking different values of f(x) & g(x) under the common interval using graphs of f and g
Q1. Can we add any two real functions?
Answer: Any two real functions having common elements in their domains can be added.
Q2.Where do we use algebra of real functions?
Answer:It is used in different mathematical concepts involving 2 or more functions like, Matrices, derivatives, integrals, continuity and differentiability, etc.
Q3.What is the algebra of functions?
Answer: Algebra of functions basically deals with the addition, subtraction, multiplication, and division of functions.