Basics of Functions and Their Graphs

Introduction to Functions

Functions are fundamental objects in mathematics that describe relationships between two sets, typically called the domain and the range. Understanding functions, their notation, and their graphical representations is essential for further study in precalculus and calculus.

• 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 usually denoted by letters such as f, g, or h. If f is a function, we write f(x) to denote the value of the function at x.

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

• Dependent Variable: The variable y = f(x) is called the dependent variable.

Example: Postal Rate Function

Consider the function that gives the postage rate for a first-class stamped letter based on its weight (up to 3.5 ounces):

• Domain: All real numbers from 0 up to and including 3.5 (ounces).

• Range: The set {0.78, 1.07, 1.36, 1.65} (dollars).

• Variables: Weight (x) is the independent variable; price (f(x)) is the dependent variable.

Example: The function can be written as a piecewise function assigning the correct price to each weight interval.

Function Notation and Evaluation

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

• Example: If f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11.

Difference Quotient

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

• Used to compute the average rate of change of a function over an interval of length h.

• Example: For f(x) = x^2, the difference quotient is:

Graphs of Functions

• The graph of a function f is the set of all points (x, f(x)) in the xy-plane.

• To graph a function, create a table of values and plot the corresponding points.

• Example: For f(x) = x^2, plot points such as (0,0), (1,1), (2,4), etc.

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.

• Application: Use the VLT to determine if a given graph represents a function.

Intercepts

• x-intercepts: Points where the graph crosses the x-axis (set f(x) = 0 and solve for x).

• y-intercepts: Points where the graph crosses the y-axis (evaluate f(0)).

Piecewise-Defined Functions

Some functions are defined by different formulas over different parts of their domain. These are called piecewise-defined functions.

• Definition: A piecewise function uses different expressions for different intervals of the domain.

• Example:

• To evaluate, determine which interval x falls into and use the corresponding formula.

Applications of Functions

• Tax Function Example: Taxes owed can be modeled as a piecewise function depending on income brackets.

• Depreciation Example: The value of a car decreasing over time can be modeled as a linear function:

for

• Here, t is the number of years since purchase, and V(t) is the car's value.

Domain of a Function

The domain of a function is the set of all real numbers for which the function's formula produces a real output.

• Unless otherwise specified, the domain is all real numbers for which the formula makes sense (e.g., no division by zero, no square roots of negative numbers).

• Example:

  • For , the domain is all real numbers except .

  • For , the domain is .

Summary Table: Key Function Concepts

Additional info: Some function names and formulas were inferred due to missing images. Examples and explanations are based on standard precalculus curriculum and the context provided.