Graphs of Functions Lesson 3

Warm Up – Perform the Operations and Simplify

Solution

Solution

Solution

Solution

Domain & Range of a Function What is the domain of the graph of the function f?

Domain & Range of a Function What is the range of the graph of the function f?

Domain & Range of a Function

Let’s look at domain and range of a function using an algebraic approach. Then, let’s check it with a graphical approach.

Find the domain and range of Algebraic Approach The expression under the radical can not be negative. Therefore, Domain Since the domain is never negative the range is the set of all nonnegative real numbers.

Find the domain and range of Graphical Approach

Increasing and Decreasing Functions

The more you know about the graph of a function, the more you know about the function itself. Consider the graph on the next slide.

Falls from x = -2 to x = 0. Is constant from x = 0 to x = 2. Rises from x = 2 to x = 4.

Increases over the entire real line. Ex: Find the open intervals on which the function is increasing, decreasing, or constant. Increases over the entire real line.

Ex: Find the open intervals on which the function is increasing, decreasing, or constant.

Ex: Find the open intervals on which the function is increasing, decreasing, or constant.

Relative Minimum and Maximum Values

The point at which a function changes its increasing, decreasing, or constant behavior are helpful in determining the relative maximum or relative minimum values of a function.

General Points – We’ll find EXACT points later……

Approximating a Relative Minimum Example: Use a GDC to approximate the relative minimum of the function given by

Put the function into the “y = “ the press zoom 6 to look at the graph. Press trace to follow the line to the lowest point.

Example Use a GDC to approximate the relative minimum and relative maximum of the function given by

Solution Relative Minimum (-0.58, -0.38)

Solution Relative Maximum (0.58, 0.38)

Step Functions and Piecewise-Defined Functions

Because of the vertical jumps, the greatest integer function is an example of a step function.

Let’s graph a Piecewise-Defined Function Sketch the graph of Notice when open dots and closed dots are used. Why?

Even and Odd Functions

Graphically

Algebraically Let’s look at the graphs again and see if this applies.

Graphically ☺ ☺

Example Determine whether each function is even, odd, or neither.

Graphical – Symmetric to Origin Algebraic

Algebraic Graphical – Symmetric to y-axis

Graphical – NOT Symmetric to origin OR y-axis. Algebraic

You Try Is the function Even, Odd, of Neither?

Solution Symmetric about the y-axis.