Introduction to Functions and Graphs

Relations and Functions

A relation is a set of ordered pairs. A function is a special type of relation where each input (x-value) has exactly one output (y-value).

• Determining if a Relation is a Function: Each x-value must correspond to only one y-value. If any x-value is paired with more than one y-value, the relation is not a function.

• Vertical Line Test: If a vertical line crosses a graph more than once, the graph does not represent a function.

• Example: The set {(1,2), (2,3), (3,4)} is a function. The set {(1,2), (1,3), (2,4)} is not a function.

Evaluating Functions

To evaluate a function, substitute the given value for x into the function's formula.

• Symbolic Evaluation: For , find : .

• Graphical Evaluation: Locate the x-value on the graph and read the corresponding y-value.

Domain and Range

The domain of a function is the set of all possible input values (x-values). The range is the set of all possible output values (y-values).

• Finding the Domain: Identify all x-values for which the function is defined (no division by zero, no square roots of negative numbers, etc.).

• Interval Notation: Example: means all real numbers less than or equal to 2.

• Set Builder Notation: Example: means all x such that x is less than or equal to 2.

• Example: For , domain is .

Increasing and Decreasing Intervals

A function is increasing on intervals where its graph rises as x increases, and decreasing where it falls.

• Finding Intervals: Examine the graph or use calculus (if known) to determine where the function rises or falls.

• Example: For , the function decreases on and increases on .

Intercepts and Zeros

• x-intercept: The point(s) where the graph crosses the x-axis ().

• y-intercept: The point where the graph crosses the y-axis ().

• Zeros of a Function: The x-values for which .

• Example: For , the zero is at .

Linear Functions and Equations

Slope and Rate of Change

The slope of a line measures its steepness and direction. The average rate of change describes how a quantity changes, on average, between two points.

• Slope Formula:

• Average Rate of Change: Same as the slope between two points on a function.

• Example: For points (1,2) and (3,6): .

Equations of Lines

• Slope-Intercept Form:

• Point-Slope Form:

• Standard Form: (not explicitly listed but commonly used)

• Finding Slope and y-Intercept: In , m is the slope, b is the y-intercept.

• Example: For , slope is 3, y-intercept is -5.

Graphing Lines

• Plot the y-intercept (b), then use the slope (rise/run) to find another point.

• Draw a straight line through the points.

Parallel and Perpendicular Lines

• Parallel Lines: Have the same slope ().

• Perpendicular Lines: Slopes are negative reciprocals ().

• Example: A line parallel to is . A line perpendicular is .

Solving Linear Equations and Inequalities

• Linear Equation: An equation of the form .

• Solving: Isolate x using algebraic operations.

• Inequality: Similar to equations, but with , , , or .

• Expressing Solutions: Use interval or set builder notation.

• Example: Solve : ; interval notation: .

Word Problems

• Distance Problems:

• Direct Variation: , where k is the constant of variation.

• Finance, Population, Cost Trends: Set up equations based on the context and solve for the unknown.

• Example: If varies directly with and when , then and .

Piecewise Functions

Graphing and Evaluating Piecewise Functions

A piecewise function is defined by different expressions for different intervals of the domain.

• Graphing: Graph each piece on its specified interval.

• Evaluating: Determine which piece applies for the given x-value, then substitute x into that expression.

• Example:

• For , ; for , .

Common Function Types and Their Graphs

Formulas to Memorize

• Slope-Intercept Form:

• Point-Slope Form:

• Slope Formula:

• Distance Formula:

• Direct Variation:

Additional info: Practice problems are referenced by section and page number for further study. The above notes cover the foundational concepts and skills needed for Test #1 in a college algebra course, focusing on functions, their properties, and linear equations.