CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB

Functions Domains and Ranges Viewing and Interpreting Graphs Even Functions and Odd functions - Symmetry Functions Defined in Pieces Absolute Value Function Composite Functions …and why Functions and graphs form the basis for understanding mathematics applications. What you’ll learn about…

Functions A rule that assigns to each element in one set a unique element in another set is called a function. A function is like a machine that assigns a unique output to every allowable input. The inputs make up the domain of the function; the outputs make up the range.

Function A function from a set D to a set R is a rule that assigns a unique element in R to each element in D. In this definition, D is the domain of the function and R is a set containing the range.

Function

Example Functions

Domains and Ranges

The domains and ranges of many real-valued functions of a real variable are intervals or combinations of intervals. The intervals may be open, closed or half-open, finite or infinite. The endpoints of an interval make up the interval’s boundary and are called boundary points. The remaining points make up the interval’s interior and are called interior points.

Domains and Ranges Closed intervals contain their boundary points. Open intervals contain no boundary points

Domains and Ranges

Graph

Example Finding Domains and Ranges [-10, 10] by [-5, 15]

Viewing and Interpreting Graphs Recognize that the graph is reasonable. See all the important characteristics of the graph. Interpret those characteristics. Recognize grapher failure. (Occurs when the graph produced by your calculator is less than precise, or even incorrect, usually due to the limitations of the screen resolution of the grapher.) Graphing with a graphing calculator requires that you develop graph viewing skills.

Viewing and Interpreting Graphs Being able to recognize that a graph is reasonable comes with experience. You need to know the basic functions, their graphs, and how changes in their equations affect the graphs. Grapher failure occurs when the graph produced by a grapher is less than precise – or even incorrect – usually due to the limitations of the screen resolution of the grapher.

Example Viewing and Interpreting Graphs [-10, 10] by [-10, 10]

Section 1.2 – Functions and Graphs Example: Use a graphing calculator to identify the domain and range, then draw the graph of the function.

Section 1.2 – Functions and Graphs Example: Use a graphing calculator to identify the domain and range, then draw the graph of the function.

Section 1.2 – Functions and Graphs You try: Use a graphing calculator to identify the domain and range, then draw the graph of the function.

Section 1.2 – Functions and Graphs You try: Use a graphing calculator to identify the domain and range, then draw the graph of the function.

Even Functions and Odd Functions-Symmetry The graphs of even and odd functions have important symmetry properties.

Even Functions and Odd Functions-Symmetry The graph of an even function is symmetric about the y-axis. A point (x,y) lies on the graph if and only if the point (-x,y) lies on the graph. (Plug in –x for x and see if it yields the same equation). The graph of an odd function is symmetric about the origin. A point (x,y) lies on the graph if and only if the point (-x,-y) lies on the graph. (Plug in –x for x and –y for y and see if it yields the same equation.)

Example Even Functions and Odd Functions-Symmetry

Section 1.2 – Functions and Gr More Examples:

Section 1.2 – Functions and Gr You try: Determine whether the following functions are even, odd or neither.

Functions Defined in Pieces While some functions are defined by single formulas, others are defined by applying different formulas to different parts of their domain. These are called piecewise functions.

Example Graphing a Piecewise Defined Function [-10, 10] by [-10, 10]

Section 1.2 – Functions and Graphs Example 2:

Absolute Value Functions Absolute Value Function can be defined as a piecewise function. The function is even, and its graph is symmetric about the y-axis

Section 1.2 – Functions and Graphs Example: Write a formula for the piecewise function whose graph consists of 2 segments, one from (0, 0) to (1, 2) and the other from (1, 0) to (2, 2)

Section 1.2 – Functions and Graphs You try: Write a formula for the piecewise function whose graph consists of 2 segments, one from (-4, -1) to (-2, 0) and the other from (-2, -1) to (0, 0)

Composite Functions

Example Composite Functions

In Review Bounded Intervals: 1. [a, b] means it is a closed interval 2. (a, b) means it is an open interval 3. [a, b) means 4. (a, b] means [ ] ab ( ) ab [ ) ab ( ] ab

In Review Unbounded Intervals 1. means 2. means 3. means 4. means 5. means the entire real number line. [ a ( a ] b ) b

In Review A function from a set D (domain) to a set R (range) is a rule that assigns a unique element in R to each element in D. The domain is the largest set of x-values for which the formula gives real y-values. The range is the set of y-values yielded by the function.

In Review Other symmetries: Symmetric about the x-axis: Plug in –y for y and see if it yields the same equation. Symmetric about the y = x line: Switch the x’s and y’s and see if it yields the same equation.

In Review Composite Functions – work from the inside out Ex:

Section 1.2 – Functions and Graphs You try: Find a formula for f(g(x)) for the following pair of functions, then find f(g(-2)).