Before starting with algebra of real functions, let’s have a look at the definition of real function. A function which has either R or one of its subsets as its range is called a real valued function. Further, if its domain is also either R or a subset of R, it is called a real function. In other words, the range of a function is a subset of real numbers, then it is a real valued function and if the domain of the function is a subset of real numbers, then it is called a real function. We can also perform algebraic operations on functions, such as addition, subtraction, multiplication, etc. In this article, algebra of real functions are defined and explained along with solved examples.
The algebraic operations that we can perform on real functions are:
- Addition of two real functions
- Subtraction of a real function from another
- Multiplication by a scalar
- Multiplication of two real functions
- Quotient of two real functions
Let’s define all these one by one along with representations and examples.
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Addition of Two Real Functions
Let f : X → R and g : X → R be any two real functions, where X ⊂ R. Then, we define (f + g): X → R by
(f + g) (x) = f (x) + g (x), for all x ∈ X
Example 1:
If f(x) = 2x^{2} and g(x) = 3x + 2 are two real functions, then find (f + g)(x).
Solution:
Given,
f(x) = 2x^{2}
g(x) = 3x + 2
Now, (f + g)(x) = f(x) + g(x) = 2x^{2} + 3x + 2
Subtraction of a Real Function From Another
Let f : X → R and g: X → R be any two real functions, where X ⊂ R. Then, we define (f – g) : X→R by
(f – g) (x) = f(x) –g (x), for all x ∈ X
Example 2:
If f(x) = x^{2} + 5x + 4 and g(x) = 17x – 5 are two real functions, then find (f – g)(x).
Solution:
Given,
f(x) = x^{2} + 5x + 4
g(x) = 17x – 5
(f – g)(x) = f(x) – g(x) = x^{2} + 5x + 4 – (17x – 5)
= x^{2} + 5x + 4 – 17x + 5
= x^{2} + (5 – 17)x + (4 + 5)
= x^{2} – 12x + 9
Therefore, the product or multiplication of given two functions is (f – g)(x) = x^{2} – 12x + 9.
Multiplication by a Scalar
Let f: X → R be a real valued function and α be a scalar. Here by scalar, we mean a real number. Then, the product α f is a function from X to R, i.e. (α f): X → R defined by
(α f) (x) = α f(x), x ∈ X
Example 3:
If f(x) = x^{2} + 2x + 1, then find (α f)(x) such that α = 5.
Solution:
Given,
f(x) = x^{2} + 2x + 1
α = 5
(α f)(x) = α f(x) = 5(x^{2} + 2x + 1) = 5x^{2} + 10x + 5
Multiplication of Two Real Functions
The product (or multiplication) of two real functions f: X → R and g: X → R is a function fg: X → R defined by
(fg) (x) = f(x) g(x), for all x ∈ X
This product of real functions is also called pointwise multiplication.
Example 4:
If f(x) = 5x – 4 and g(x) = x^{2} – 9 are two real functions, then find (fg)(x).
Solution:
Given,
f(x) = 5x – 4
g(x) = x^{2} – 9
Now,
(fg) (x) = g(x) g(x)
= (5x – 4)(x^{2} – 9)
= (5x)(x^{2}) + (5x)(-9) + (-4)(x^{2}) + (-4)(-9)
= 5x^{3} – 45x – 4x^{2} + 36
Therefore, (fg)(x) = 5x^{3} – 4x^{2} – 45x + 36
Quotient of Two Real Functions
Let f and g be two real functions defined from X to R, i.e. f: X → R and g: X → R, where X ⊂ R. The quotient of f by g denoted by \(\frac{f}{g}\) is a function defined by,
\(\left ( \frac{f}{g} \right )(x)=\frac{f(x)}{g(x)}\), provided g(x) ≠ 0 and x ∈ X
Example 5:
If f(x) = x^{2} and g(x) = √x are two real functions such that x is non-negative real number, then find \(\left ( \frac{f}{g} \right )(x)\).
Solution:
Given,
f(x) = x^{2}
g(x) = √x
Now,
\(\left ( \frac{f}{g} \right )(x)=\frac{f(x)}{g(x)} = \frac{x^2}{\sqrt{x}} = \frac{x^2}{x^{\frac{1}{2}}}\)= x^{2} . x^{-1/2}
= x^{3/2}; x is a non-negative real number.