Algebraic Functions: Definition, Types, Operations in Mathematics

Algebraic functions are functions that contain algebraic expressions involving variables. Learning how to perform operations on algebraic functions is an important skill in algebra.

In this article, we will discuss the basic rules of algebraic functions.

What are Algebraic Functions?

An algebraic function is a function that contains variables and algebraic operations. These operations include addition, subtraction, multiplication, division, and exponentiation.

The general form of an algebraic function is:

f(x) = P(x) / Q(x)

• f(x) is the function itself.
• P(x) and Q(x) are just names for polynomials (which are expressions made up of variables and constants).
• The x in the equation represents the input values.

Algebraic Function Examples

• f(x) = 3x + 5
• g(x) = x³ – 4x + 7
• h(x) = (3x + 1) / (2x – 1)
• k(x) = x³

Types of Algebraic Functions

There are three types of Algebraic Functions.

• Polynomial Functions

• Rational Functions

• Power Functions

Polynomial Functions

Polynomial functions made up with variables and constants, all combined with the usual addition, subtraction, multiplication, and division.

For example,

• f(x) = 3x + 7
• f(x) = x² – 2x + 5
• f(x) = x³ – 7x + 7
• f(x) = x⁴ – 5x^2 + 2x – 8

Rational Functions

Rational functions are a bit like fractions but with a variable in the denominator (and sometimes in the numerator).

They look like f(x) = p(x)/q(x), where p(x) and q(x) are polynomials.

For example,

• (x – 1) / (3x + 2)
• (5x – 7) / (x² – 7x + 9)
• (4x² + 1) / (x + 2)

Power Functions

The power functions are of the form f(x) = k x^a, where ‘k’ and ‘a’ are any real numbers. The exponent ‘a’ can be an integer or a fraction.

For example,

• x²
• x^-1 (also known as a reciprocal function)
• √(x – 2) (which means the square root of x – 2)
• 3√(x – 3)

Adding and Subtracting Algebraic Functions

To add or subtract two algebraic functions, we combine like terms:

f(x) = 3x + 5
g(x) = 2x – 1

f(x) + g(x) = (3x + 5) + (2x – 1)
= 5x + 4

Multiplying Algebraic Functions

To multiply two algebraic functions, we distribute and combine like terms:

f(x) = 3x + 5
g(x) = 2x – 1

f(x) * g(x) = (3x + 5)(2x – 1)
= 6x² + x – 5

Dividing Algebraic Functions

To divide two algebraic functions, we divide the numerator by the denominator as we would with regular numbers:

f(x) = 6x² + x – 5
g(x) = 2x – 1

f(x) / g(x) = (6x² + x – 5) / (2x – 1)

Solved Examples Algebraic Functions

Example

Given f(x) = 3x + 2 and g(x) = x – 4, find f(x) + g(x).

Solution

f(x) = 3x + 2
g(x) = x – 4
f(x) + g(x) = (3x + 2) + (x – 4)
= 4x – 2

Example

Given f(x) = 2x^2 – x and g(x) = 3x + 5, find f(x) * g(x).

Solution

f(x) = 2x² – x
g(x) = 3x + 5
f(x) * g(x) = (2x² – x)(3x + 5)
= 6x³ + 7x² – 15x – 5

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