Basics of Functions and Their Graphs

Introduction

This section introduces the foundational concepts of functions, their notation, graphical representation, and applications. Understanding functions is essential for all further study in precalculus and calculus, as they provide a framework for modeling relationships between variables.

Functions: Notation and Definitions

• Function: A function is a rule that assigns to each element x in a set called the domain exactly one element y in a set called the range.

• Notation: Functions are typically denoted by letters such as f, g, or h. If f is a function, we write to denote the value of the function at x.

• Independent Variable: The variable x (input) is called the independent variable.

• Dependent Variable: The variable y (output), often written as , is called the dependent variable.

• Domain: The set of all possible input values (x) for which the function is defined.

• Range: The set of all possible output values (y or ) that the function can produce.

Example: Postal Rate Function

The cost of mailing a first-class stamped letter (as of July 13, 2025) is a function of its weight. The function can be represented by the following table:

This is an example of a piecewise-defined function, where the output (price) depends on the interval in which the input (weight) falls.

Function Rules and Formulas

• Some functions are defined by explicit formulas, such as .

• To evaluate a function at a specific value, substitute the value into the formula.

Example: Evaluating a Function

• Given , find:

Difference Quotient

The difference quotient is a fundamental concept for understanding rates of change and is defined as:

• This expression is used extensively in calculus to define the derivative.

Example: Calculating the Difference Quotient

• Given , the difference quotient is:

Graphs of Functions

• The graph of a function is the set of all points in the coordinate plane.

• To graph a function, plot several ordered pairs and connect them appropriately.

Vertical Line Test (VLT)

• A set of points in the plane is the graph of a function if and only if no vertical line intersects the graph at more than one point.

• This test helps determine whether a given graph represents a function.

Example: Using the Vertical Line Test

• If a graph passes the VLT, it is a function. For example, the graph of passes the VLT, but the graph of a circle does not.

Intercepts

• x-intercepts: Points where the graph crosses the x-axis ().

• y-intercepts: Points where the graph crosses the y-axis ().

Piecewise-Defined Functions

• A piecewise-defined function is defined by different formulas on different parts of its domain.

Example: Piecewise Function

• To evaluate , use the first formula: .

• To evaluate , use the second formula: .

Applications of Functions

• Functions are used to model real-world situations, such as tax calculations or depreciation of assets.

Example: Tax Function (Piecewise)

• Suppose the tax owed is defined by a piecewise function depending on income .

• To find , substitute into the appropriate formula based on the income bracket.

Example: Car Depreciation

• The value of a car purchased for that decreases by per year for the first seven years is:

• , for

• To find , compute

Domain of a Function

• Unless otherwise specified, the domain of a function is all real numbers for which the formula makes sense.

• Common restrictions include:

  • Denominators cannot be zero.

  • Even roots (e.g., square roots) require non-negative radicands.

Example: Finding the Domain

• For , the domain is all real numbers except .

• For , the domain is .

• For , the domain is .

Additional info: Some function names and formulas were inferred based on standard precalculus examples, as the original file referenced images for notation and specific function definitions.