Identifying and Classifying Different Types of Functions

In Grade 9 Pre-Calculus, it’s important to understand the different types of functions.

Functions can be grouped into several main categories based on their features. Here are the main types:

1. Linear Functions:

   • General Form: ( f(x) = mx + b )
     Here, ( m ) is the slope (how steep the line is) and ( b ) is where the line crosses the y-axis (the y-intercept).
   • Characteristics: The graph is a straight line. The degree is 1.
     Example: ( f(x) = 2x + 3 ).

2. Quadratic Functions:

   • General Form: ( f(x) = ax^2 + bx + c )
     where ( a ) is not zero.
   • Characteristics: The graph makes a U-shape called a parabola. The degree is 2.
     Example: ( f(x) = x^2 - 4x + 4 ).

3. Polynomial Functions:

   • General Form: ( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0 )
     where ( n ) is a whole number (non-negative).
   • Characteristics: Depending on the degree, the graph can take many shapes.
     Example: For cubic functions, ( f(x) = x^3 - 3x + 2 ).

4. Rational Functions:

   • General Form: ( f(x) = \frac{p(x)}{q(x)} )
     Here, ( p(x) ) and ( q(x) ) are polynomials.
   • Characteristics: These graphs might have lines that they approach but never touch, called asymptotes. The degree of ( p ) can be less than, greater than, or equal to the degree of ( q ).
     Example: ( f(x) = \frac{x^2 - 1}{x + 2} ).

5. Exponential Functions:

   • General Form: ( f(x) = a \cdot b^x )
     where ( b ) is greater than 0 and not equal to 1.
   • Characteristics: These functions grow or shrink very quickly.
     Examples: ( f(x) = 2^x ) or ( f(x) = 3e^{0.5x} ).

To identify and classify these functions easily, pay attention to their general forms, their graphs, and how they behave. Look for where the graphs cross the axes, any asymptotes, and their end behavior.