Introduction to Functions

This document provides an introduction to functions, explaining how to determine if a rule describes a valid function and introduces key mathematical terms associated with functions. It emphasizes the importance of practice exercises to master these concepts.

Learning Objectives

• Recognize when a rule describes a valid function.
• Plot the graph of a function.
• Identify a suitable domain for a function and find the corresponding range.

Key Concepts

1. What is a Function?

• A function is a rule mapping a number to another unique number.
• Example: If a function adds 3 to any number, then:
  • For input 2: ( f(2) = 2 + 3 = 5 )
  • For input 8: ( f(8) = 8 + 3 = 11 )
  • Mathematically represented as ( f(x) = x + 3 ).
• Terminology:
  • The input is called the independent variable or argument.
  • The output is the dependent variable.

2. Plotting the Graph of a Function

• To plot a function graph, calculate the output for various inputs.
• Example: For ( f(x) = 3x^2 - 4 ):
  • ( f(0) = -4 )
  • ( f(1) = -1 )
  • ( f(2) = 8 )
• Use a table of values to plot the function graph.
• The graph helps determine output corresponding to an argument.

3. When is a Function Valid?

• A function must map each argument to a unique output.
• Visual Check: If a vertical line crosses the graph more than once, it's not a valid function.
• Domain and Range:
  • Domain: Set of possible inputs.
  • Range: Set of corresponding outputs.
  • Example of domain restriction: ( f(x) = \sqrt{x} ) where ( x \geq 0 ).

4. Some Further Examples

• Validity check involves ensuring unique output per input and assessing domain restrictions.
• Example 1: ( f(x) = 2x^2 - 3x + 5 )
  • Check outputs for specific inputs to validate function.
• Example 2: ( f(x) = \frac{1}{x} )
  • Domain restriction: ( x \neq 0 ) because division by zero is undefined.
  • Approaching zero from different directions (left/right) yields different outputs.
• Example 3: ( f(x) = \frac{1}{(x-2)^2} )
  • Domain restriction: ( x \neq 2 ) due to division by zero.
  • The graph has an asymptote at ( x = 2 ).

Exercises

1. Consider the function ( f(x) = 2x^2 + 5x - 3 )
   • Identify the argument and dependent variable.
   • Plot the function and calculate specific values.
   • Determine domain and range.
2. Consider ( f(x) = \frac{1}{(x-3)^2} )
   • Plot graph, determine domain and range.
3. Investigate ( f(x) = 3x )
   • Assumptions for validity, plot graph, and determine domain/range.
4. For ( f(x) = \frac{1}{x} )
   • Plot graph, and study behavior as ( x ) approaches zero.
5. Determine domain and range for listed functions and pair with similar domain/range functions.

Conclusion

Understanding functions involves recognizing valid functions, plotting graphs, and defining domains and ranges. Through examples and exercises, these concepts are reinforced, ensuring a comprehensive grasp of function fundamentals.