# 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) = x
^{3}– 4x + 7 - h(x) = (3x + 1) / (2x – 1)
- k(x) = x
^{3}

**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) = x
^{2}– 2x + 5 - f(x) = x
^{3}– 7x + 7 - f(x) = x
^{4}– 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) / (x
^{2}– 7x + 9) - (4x
^{2}+ 1) / (x + 2)

**Power Functions**

The **power function**s 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,

- x
^{2} - x
^{-1}(also known as a reciprocal function) - âˆš(x – 2) (which means the square root of x – 2)
- 3âˆš(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)

= 6x^{2} + 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) = 6x^{2} + x – 5

g(x) = 2x – 1

f(x) / g(x) = (6x^{2} + 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) = 2x^{2} – x

g(x) = 3x + 5

f(x) * g(x) = (2x^{2} – x)(3x + 5)

= 6x^{3} + 7x^{2} – 15x – 5

**FAQs**

**What is the difference between an independent variable and dependent variable?**

The independent variable is the variable x in the function. The dependent variable is the output value we get after substituting a number for x.

**What are some examples of algebraic functions?**

Examples include:

f(x) = 2x + 1

g(x) = x^{2} – 3x + 5

h(x) = (x+1) / (x-2)

**How do you evaluate an algebraic function?**

To evaluate, substitute the given value of x into the function and simplify. For example, given f(x) = 2x + 1, f(3) = 2(3) + 1 = 7.

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