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Algebraic Function
An algebraic function, as its name suggests, is a function that is made up of only algebraic operations. There are different types of functions we study in math. The most common functions are:
 Algebraic functions
 Trigonometric functions
 Logarithmic functions
 Exponential functions
Let us learn more about algebraic functions along with their types and examples.
1.  What is an Algebraic Function? 
2.  Types of Algebraic Functions 
3.  Algebraic Functions Graphs 
4.  FAQs on Algebraic Functions 
What is an Algebraic Function?
An algebraic function is a function that involves only algebraic operations. These operations include addition, subtraction, multiplication, division, and exponentiation. Based on this definition, let us see some examples of algebraic functions and nonalgebraic functions.
Algebraic Function Examples
Here are some examples of algebraic functions. Note that algebraic functions should include only the operations, +, , ×, ÷, integer and rational exponents. These notations result in algebraic functions such as a polynomial function, cubic function, quadratic function, linear function, and is based on the degree of the equations involved.
 f(x) = x^{2}  5x + 7
 g(x) = √x
 h(x) = (3x + 1) / (2x  1)
 k(x) = x^{3}
Identifying Algebraic Functions
If a function includes only the abovementioned operations (+, , ×, ÷, exponents (also roots)), then we can say that it is an algebraic function. Let us have a look at nonalgebraic functions as well to avoid confusion.
NonAlgebraic Function Examples
Nonalgebraic functions include trigonometric functions, logarithmic functions, absolute value functions, exponential functions, etc. Here are some examples.
 f(x) = sin (3x + 2)
 g(x) = log x
 h(x) = 3^{x}
Types of Algebraic Functions
Based upon the above examples, you might already have got an idea to segregate the types of algebraic functions. Here are the main types.
 Polynomial Functions
 Rational Functions
 Power Functions
Let us see more examples of each of these types.
Polynomial Functions
The polynomial functions (which are one type of algebraic functions) are functions whose definition is a polynomial. The polynomial functions include linear function, quadratic function, cubic function, biquadratic function, quintic function, etc. Here are some examples.
 f(x) = 3x + 7 (linear function)
 f(x) = x^{2}  2x + 5 (quadratic function)
 f(x) = x^{3}  7x + 7 (cubic function)
 f(x) = x^{4}  5x^{2} + 2x  8 (biquadratic function)
 f(x) = x^{5}  7x + 3 (quintic function)
The domain of all polynomial functions is the set of all real numbers and the range depends upon the yvalues that the graph covers. To know more about the polynomial functions, click here.
Rational Functions
The rational functions (which are one type of algebraic functions) are functions whose definition involves a fraction with variable in the denominator (they may have variable in the numerator as well). i.e., they are of the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials in x. Here are some examples:
 f(x) = (x  1) / (3x + 2)
 f(x) = (5x  7) / (x^{2}  7x + 9)
 f(x) = (4x^{2} + 1) / (x + 2)
To find the domain of rational functions, we use the rule denominator ≠ 0 and to find the range, we solve the function for x and then apply the same rule denominator ≠ 0. To know more about the rational functions, click here.
Power Functions
The power functions are of the form f(x) = k x^{a}, where 'k' and 'a' are any real numbers. Since 'a' is a real number, the exponent can be either an integer or a rational number. Here are some examples.
 f(x) = x^{2}
 f(x) = x^{1 }(reciprocal function)
 f(x) = √(x  2) = (x  2)^{1/2}
 f(x) = \(\sqrt[3]{x3 }\) = (x  3)^{1/3}
The domain of all power functions may not be the same. That depends upon the xvalues where the function is defined. The range of power functions depends upon the yvalues that the graph would cover.
Algebraic Functions Graphs
The graphs of all algebraic functions are NOT the same. It depends upon the equation of the function. The general procedure to graph any y = f(x) is:
 Find the xintercepts (by setting y = 0)
 Find the yintercepts (by setting x = 0)
 Find all asymptotes and plot them.
 Find the critical points and inflection points.
 Find some extra points in between every two xintercepts and in between every two asymptotes.
 Plot all these points and join them curves by taking care of the asymptotes.
For more information on graphing functions, click here.
Important Notes on Algebraic Function
 Algebraic functions include only algebraic operations.
 The algebraic operations are addition, subtraction, multiplication, division, powers, and roots.
 Any function that has a log, ln, trigonometric functions, inverse trigonometric functions, or variable in the exponent is NOT an algebraic function.
 The domain and range of any algebraic function can be found by graphing it on a graphing calculator and seeing the xvalues and the yvalues respectively that the graph would cover.
☛ Related Topics:
Algebraic Functions Examples

Example 1: Which of the following are algebraic functions? a) f(x) = ln (x  5) b) f(x) = (x^{2} + 2)^{1/2} c) f(x) = sin (x^{3}).
Solution:
We know that algebraic functions should involve only algebraic operations.
(a) f(x) = ln (x  5) is NOT algebraic as it is a logarithmic function.
(b) f(x) = (x^{2} + 2)^{1/2} is algebraic.
(c) f(x) = sin (x^{3}) is NOT algebraic as it is a trigonometric function.
Answer: Only (b).

Example 2: Find the domain and range of the algebraic function f(x) = x^{2}  4x + 5.
Solution:
Since f(x) is a polynomial, its domain is the set of all real numbers.
Since the given function is quadratic, its range depends upon the ycoordinate of its vertex. To find its vertex, we will complete the square.
f(x) = x^{2}  4x + 5
Adding and subtracting 4,
f(x) = x^{2}  4x + (4  4) + 5
f(x) = (x^{2}  4x + 4)  4 + 5
f(x) = (x  2)^{2} + 1Comparing this with vertex form, we get its vertex to be (2, 1).
Also, the coefficient of x^{2} is 1 which is positive and so the parabola opens up.
So the range of the function is { y ∈ R  y ≥ 1}.
Answer: Domain = R and Range = { y ∈ R  y ≥ 1}.

Example 3: Check the domain and range of the algebraic function from Example 2 graphically.
Solution:
Graph the function f(x) = x^{2}  4x + 5 either using the graphing calculator or using any graphing tool.
From the graph:
 Domain = Set of all xvalues covered by the graph = (∞, ∞) = R
 Range = Set of all yvalues covered by the graph = { y ∈ R  y ≥ 1}
Answer: The domain and range of the function in Example 2 are verified graphically.
FAQs on Algebraic Functions
What is Algebraic Function Definition?
An algebraic function is a type of functions that is formed only with the following operations:
 Addition
 Subtraction
 Multiplication
 Division
 Exponents (Integer or rational)
What is Algebraic Functions List?
Algebraic functions are constructed only using algebraic operations. There are 3 types in this:
 Polynomial functions
 Power functions
 Rational functions
What are Algebraic Function Types?
Algebraic functions use only the operations: +, , ×, ÷, and exponent (any rational exponent) and they are of 3 types:
 Rational functions
 Polynomial functions
 Power functions
Is log x an Algebraic Function?
No, log x is NOT algebraic. It is a logarithmic function.
How to Identify Algebraic Functions?
An algebraic function should include only the following operations:
 +
 
 ×
 ÷
 Powers like x^{2}, (x  1)^{1/2}, etc
If we have anything apart from these operations then the function is NOT algebraic.
Is Sin x an Algebraic Function?
No, sin x is NOT algebraic. It is a trigonometric function.
Which Functions are NOT Algebraic?
The functions that involve anything other than +, , ×, ÷, and exponent are NOT algebraic. Some examples of functions that are not algebraic are:
 f(x) = sin (x + 2)
 f(x) = ln (x^{3}/3), etc.
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