Functions are broadly classified as Algebraic functions and Transcendental functions. You are already familiar with some of the algebraic functions like Linear, Quadratic and higher order Polynomial functions.

Polynomials form the building blocks for other advanced algebraic functions. Algebraic functions are formed by allowing basic algebraic operations on Polynomials.

Algebraic functions are constructed from polynomials using algebraic operations like addition, subtraction, multiplication division and square roots.

Examples:
1. f(x) = $\sqrt{x^{2}+4}$
f(x) is formed taking square root of the Polynomial x2 + 4.

2. g(x) = $\frac{x^{2}+5x+6}{x-4}$
g(x) is a rational function of the form $\frac{p(x)}{q(x)}$ where p(x) and q(x) are polynomials.

3. h(x) = $\frac{x^{3}-2x^{2}}{x+\sqrt{x}}$
h(x) is formed by dividing the polynomial $x^{3}-2x^{2}$ by another algebraic function $x+\sqrt{x}$.
There are three common types of algebraic functions:
  1. Polynomials: Polynomials which are the building blocks of algebraic functions are themselves algebraic function.
  2. Rational functions: Rational functions are formed by dividing one polynomial by another. In general a rational function is of the form $\frac{p(x)}{q(x)}$ where p(x) and q(x) are polynomials.
  3. Radical Functions: The Radical functions are formed by taking roots of the polynomials.
A rational function is an algebraic function expressed as a quotient.

f(x) = $\frac{p(x)}{q(x)}$, where p(x) and q(x) are polynomials. The domain of f(x) consists of all values of x such that q(x) $\neq$ 0.

The reciprocal function f(x) = $\frac{1}{x}$ is the simplest rational function with domain x $\neq$ 0. This is also known as the parent function for the family of all rational functions.

Graphs of Polynomials
When compared to the graphs of Polynomials, additional behavioral patterns can be observed in the graphs of rational functions, which are discontinuity and asymptotes.
Breaks are discontinuity seen in the graphs of rational functions at values excluded from the domain of the function.
If the function is given in its simplest form, vertical asymptotes occur at the points where the function is not defined. A vertical asymptote is a vertical line which appears to change the course and position of the graph.

If the function given is not in simplest form, that is when p(x) and q(x) have common factors, a point discontinuity is caused at the excluded value. The graph of the function has a break at the point and it is said to have a hole at the point.
The rational functions have horizontal asymptotes at excluded values in the range of the function. The graph of rational function appears to coincide with a horizontal line (which is the horizontal asymptote) as the value of x becomes larger in either or one of positive and negative directions.

Rational Algebraic Functions
    
The graph of rational function f(x) = $\frac{x+2}{x^{2}-x-6}$ is shown on the left.
The graph is undefined at x = 3 and x = -2. The graph has the vertical
asymptote at x = 3 and a hole at x = -2. Note here (x + 2) is a common factor
of the numerator and the denominator.

The graph also has the horizontal asymptote at y = 0, the x axis.


The graph of a rational function can also have a slant asymptote when the degree of the numerator polynomial is greater than that of the denominator.
Rules for common algebraic functions are given along with an example graph.

1. Quadratic Functions:
  1. Determine the vertex and the y intercept.
  2. Find the x intercepts if it is easier to find the zeros of the quadratic by factoring.
  3. Find few points symmetric to the vertex, taking x values symmetric to the x coordinate of the vertex.
  4. Plot the vertex, intercepts and the points found.
  5. Make a neat sketch of the graph joining the points smoothly.

Quadratic Function

2. Polynomials of Higher degree

  1. Find the y intercept of the polynomial
  2. Find the zeros of polynomials if they can be easily determined using synthetic division and factoring.
  3. Determine the end behavior of the polynomial using the leading term.
  4. Find few points to plot using the function definition.
  5. Roughly determine the turning points using trial values.
  6. Plot the intercepts, turning points and graph, in accordance to the end behavior of the polynomial.

Polynomials


3. Rational functions:
  1. Find the x and y intercepts if they exist.
  2. Find the vertical asymptotes and holes.
  3. Find the horizontal or slant asymptote.
  4. Find few points around the critical values.
  5. Draw the asymptotes, plot the intercepts and points found.
  6. Sketch the graph smoothly in tune with the behavioral elements.

4. Radical Functions: 

  1. Determine the base line using the minimum or maximum function value.
  2. Find few points taking convenient x values, so that the corresponding y values are rational numbers.
  3. Plot the points and draw the graph smoothly.
Radical

Solved Examples

Question 1: Find the domain and range of  the rational function f(x) = $\frac{x^{2}+7x+6}{(x^2-  4}$, and also,
  1. x and y intercepts
  2. Vertical asymptotes
  3. Horizontal asymptote.

Solution:
 

The function is not defined when x2 - 4 = 0   ⇒ (x + 2)(x - 2) = 0
That is the function is not defined when x = -2 and x = 2.
The Domain of the function is {x | x $\neq$ -2, 2}
The excluded value for y is y = 1 applying infinite limits .
Hence the range of the function is {y | y $\neq$ 1}

1. The x intercept of the function can be found by setting  f(x) = 0

$\frac{x^{2}+7x+6}{x^2 - 4}$ = 0

x2 + 7x + 6 = 0     ⇒  (x + 6)(x + 1) = 0   ⇒  x = - 6, - 1
The x intercepts are x = - 6 and x = - 1.

The y intercept is given by f(0) and f(0) = $\frac{6}{-4}$ = -1.5

2. The function has Vertical asymptotes at the points where the function is not defined.
Thus the vertical asymptotes are x = -2 and x = 2.

3. As the degree of the polynomials in the numerator and denominator are same, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficients in both numerator and denominator = 1.

Their ratio = $\frac{1}{1}$ = 1.

Horizontal asymptote is hence y = 1.
 

Question 2: Find the domain and range of the quadratic function f(x) = -(x - 3)2 + 5
Solution:
 
The quadratic function is defined for all values of x.
   Hence Domain is all real numbers.

   The vertex of the graph is (3, 5). Since the leading term is negative, the graph bends U down and the maximum function value is reached at the vertex.
    Maximum function value = 5    Thus range of f(x) = $(-\infty$ ,5]