PreCalculus 1st Semester 1.3 Graphs of Functions PreCalculus 1st Semester

Objectives: Find domains and ranges of functions and use the Vertical Line Test for functions. Determine intervals on which functions are increasing, decreasing , or constant. Determine relative maximum and minimum values of functions. Identify and graph step functions and other piecewise- defined functions. Identify even and odd functions.

Interval Notation Use intervals to write continuous, infinite sets with the following guidelines: Brackets [] – indicates that the endpoint is included. Never use brackets with infinity. Parenthesis () – indicates that the endpoint is not included.

The Graph of A Function The graph of a function f is the collection of ordered pairs (x, f(x)) such that x is the domain of f. x = the directed distance from the y-axis. f(x) = the directed distance from the x-axis To find domain: movement left to right To find range: movement bottom to top

Example 1: Find the domain and range of the following functions: (b)

Vertical Line Test A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point. Not a function function

Increasing and Decreasing Functions Trace the graph from left to right (x-axis) to indicate when the function increases, decreases or is constant. Only look at the movement on the x-axis! Always use parentheses for increasing and decreasing! Decreasing (−∞, −3) Constant (−3, 4) Decreasing (4, ∞)

Example 2: (b) (a) Decreasing (−∞, 4) Decreasing (−∞, 0) Increasing (0, 2) Increasing (4, ∞) Decreasing (2, ∞)

Example 2: (c) Decreasing (−∞, −1) Constant (−1, 1) Decreasing (1, ∞)

Minimum and Maximum Values Points at which the function changes direction from increasing to decreasing (or decreasing to increasing) These points are called critical points. Relative Maximum – a high point on the visible graph. Absolute Maximum – the highest point the graph ever reaches. Relative Minimum – a low point on the visible graph. Absolute Minimum – the lowest point the graph ever reaches.

Example 3:

Example 3:

Greatest Integer Function

Example 4: 1 2 3 -4 -3 2

Graphing Piecewise Functions 1)Rewrite in slope – intercept form 2) Graph using slope and y-intercept and/or choose points 3) Open dot < or > 4) Closed dot

Example 5: Graph

Even and Odd Functions A function whose graph is symmetric with respect to the y-axis is an even function. A function whose graph is symmetric with respect to the origin is an odd function. A graph that is symmetric to the x-axis is not the graph of a function.

Even and Odd Functions Three ways to determine if a function is even or odd: Algebraically Graphically Examination

Algebraically:

Example 6:

Graphically: Even Even Odd Odd Not a function

Examination: