Lecture Notes: Linear Functions

Definition of a Linear Function

• A linear function is a function that represents a straight line on a coordinate plane.
• General form: f(x) = mx + b
  • m: slope of the line
  • b: y-intercept of the line
  • x: independent variable
  • y or f(x): dependent variable

Characteristics of Linear Functions

• Represents a line, hence the graph is a straight line.
• Involves only algebraic operations.
• Parent linear function: f(x) = x, which passes through the origin.

Linear Function Equation Examples

• f(x) = 3x - 2
• f(x) = -5x - 0.5
• f(x) = 3

Real-Life Applications

1. Movie streaming service fees: f(x) = 0.35x + 4.50
2. T-shirt printing fees: f(x) = 7x + 50
3. Linear programming for cost minimization or profit maximization.

Finding a Linear Function

• Use slope-intercept form or point-slope form.
• Example: Finding the linear function using points (-1, 15) and (2, 27)
  • Slope (m) calculation: m = 4
  • Linear equation: f(x) = 4x + 19

Identifying a Linear Function

• Graph: A line indicates a linear function.
• Equation: Form f(x) = mx + b.
• Table: Constant ratio of differences in y-values to x-values.

Graphing a Linear Function

• Using Two Points: Choose random x-values, find y, plot and join.
• Using Slope and Y-Intercept:
  • Plot y-intercept.
  • Use slope to find another point, connect and extend line.

Domain and Range

• Generally, both domain and range for linear functions are all real numbers (R).
• Special case: Horizontal line f(x) = b has domain = R, range = {b}.

Inverse of a Linear Function

• Process to find inverse:
  1. Replace f(x) with y.
  2. Swap x and y, solve for y.
  3. Replace y with inverse function notation.
• Inverse functions are symmetric about y = x.

Piecewise Linear Function

• Defined in multiple ways over its domain.
• Example includes different linear equations for different domain sections.

Important Notes

• Linear functions graph as lines.
• Parallel lines have equal slopes.
• Perpendicular lines have a slope product of -1.
• Vertical lines are not linear functions.

Additional Examples

1. Celsius to Fahrenheit conversion: F = (9/5)C + 32
2. Car rental costs over days: C(x) = 30x + 20

Practice and FAQ

• Questions on identifying linear functions.
• FAQs cover definitions, graphing, and differences in linear vs. nonlinear functions.

This guide provides a comprehensive understanding of linear functions, their characteristics, applications, and graphical representations, useful for students in calculus and algebra courses.