Algebra of Real Functions

Algebra of Real Functions

Algebra of real functions refers to applying algebraic operations to functions that have real numbers as their domain and range. These operations include addition, subtraction, multiplication, and division, which are used to simplify expressions and solve equations involving real functions.

Here are some examples of algebraic operations on real functions:

Addition of Two Real Functions

If f(x) and g(x) are two real functions, then their sum is given by:

(f + g)(x) = f(x) + g(x)

For example, if f(x) = x2 + 2x + 3 and g(x) = 2x - 5,

then (f + g)(x) = x2+ 4x - 2

Subtraction of a Real Function From Another

If f(x) and g(x) are two real functions, then their difference is given by:

(f - g)(x) = f(x) - g(x)

For example, if f(x) = x2 + 2x + 3 and g(x) = 2x - 5, then (f - g)(x) = x2 + x + 8.

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 

Multiplication of Two Real Functions

If f(x) and g(x) are two real functions, then their product is given by:

(f * g)(x) = f(x) * g(x)

For example, if f(x) = x2 - 3x + 2 and g(x) = 2x + 1, then (f * g)(x) = 2x3 - 4x2 - 7x + 2.

Quotient of Two Real Functions

If f(x) and g(x) are two real functions, and g(x) is not zero, then their quotient is given by:

(f / g)(x) = f(x) / g(x),  provided g(x) ≠ 0 and x X

For example, if f(x) = x2 + 2x + 1 and g(x) = x + 1, then (f / g)(x) = x + 1.

These algebraic operations can be used to simplify expressions and solve equations involving real functions. For example, consider the following equation:

x2 - 3x + 2 = 0

We can use algebraic techniques to factor the expression and solve for the roots:

x2 - 3x + 2 = (x - 1)(x - 2) = 0

Therefore, x = 1 or x = 2.