Algebraic Function

An algebraic function is a function f(x) that satisfies p(x,f(x))=0, where p(x,y) is an integer-coefficient polynomial in x and y. Algebraic functions are functions that can be created using just a limited number of elementary operations, as well as the inverses of functions that can be constructed in this way. In this article, we will learn more about algebraic functions with some examples.

Definition of Algebraic Function

An operation that can only be performed using algebraic concepts is referred to as an algebraic function. The operations of addition, subtraction, multiplication, division, and exponentiation are all included in this category. 

Examples of Algebraic Function

Here are some algebraic function examples. Only the operations +, -,,, integer, and rational exponents should be included in algebraic functions. Based on the degree of the equations involved, these notations result in algebraic functions such as a polynomial function, cubic function, quadratic function, and linear function.

f(x) = x² – 2x + 1

g(x) = 8x + 3/4x – 2

k(x) = x⁴

Types of Algebraic Functions

You may have already come up with a way to categorise the many forms of algebraic functions. The most common types are discussed here:

1. Polynomial Function

Polynomial functions (another form of algebraic function) are functions with a polynomial definition. Linear functions, quadratic functions, cubic functions, biquadratic functions, and quintic functions are examples of polynomial functions. Here are a few illustrations.

f(x) = 5x + 2 [linear]

f(x) = 3x² – 4x + 2 [quadratic]

f(x) = 4x³ – 1 [cubic]

f(x) = 2x⁴ + 6x – 5 [bi-quadratic]

f(x) = 7x⁵ – 2x² + 3 [quintic]

The set of all real numbers is the domain of all polynomial functions, and the range is determined by the y-values that the graph covers.

2. Rational Function

The rational functions (which are a form of algebraic function) are functions that have a fraction in the denominator and a variable in the numerator (they may have variables in the numerator as well). Specifically, they have the form f(x) = p(x)/q(x), with p(x) and q(x) being polynomials in x. Some instances are as follows:

g(x) = (3x² – 2x + 1)/(2x – 1)

g(x) = (5x – 2)/(3x + 1)

g(x) = (7x³ + 2)/(x + 2)

We utilise the rule denominator 0 to identify the domain of rational functions, and we solve the function for x and then apply the same rule denominator 0 to obtain the range.

3. Power Function

Power functions have the form f(x) = k x^a, where ‘k’ and ‘a’ can be any real values. The exponent can be either an integer or a rational number because ‘a’ is a real number. Here are a few illustrations.

k(x) = x³

k(x) = x^-1

k(x) = √(x – 2)

Identification of Algebraic Functions

We can call a function algebraic if it simply contains the above-mentioned operations (+, -, *, /) exponents (including roots), and so on.

Algebraic Functions Graph

All algebraic functions do not have the same graph. It is determined by the function’s equation. The following is the usual approach for graphing any y = f(x):

• Set y = 0 to find the x-intercepts

• Set x = 0 to find the y-intercepts

• Locate and plot all asymptotes

• Determine important and inflection points

• There should be some extra points between every two x-intercepts and asymptotes

• Plot all of these points and connect them with curves, paying attention to the asymptotes

Facts of Algebraic Function

1. Algebraic functions don’t comprise anything but algebraic methods however.

2. The mathematical operations that make up algebra include addition, subtraction, multiplication, division, powers, and roots. 

3. A logarithmic function, or a natural logarithmic function, or a trigonometric function, or an inverse trigonometric function, or a variable in the exponent is an example of an algebraic function. 

4. The domain and range of any algebraic function may be found by graphing the function on a graphing calculator and examining the x- and y-values that the graph would cover, respectively. This method is applicable to any algebraic function. 

Conclusion

After gaining a knowledge of the notion of algebraic functions, one thing that is abundantly evident is that algebraic expressions play an important role not just in the mathematics curriculum but also in daily life. In order for students to make progress and achieve success in mathematics, they need to have the ability to not only read and write expressions but also calculate and manipulate algebraic expressions. Because of the basic theorem of Galois theory, it is not necessary for algebraic functions to be represented by radicals. This is an oversimplification of the situation. Now that we have that out of the way, let’s go on to learning more about algebraic functions.