Functions Lesson 1.3

Today we look at the mathematical way of talking about the "rule" The Magic Box 1 -3 4 3 3 1 Consider a box that receives numbers in the top And alters them somehow and sends a (usually) different number out the spout Today we look at the mathematical way of talking about the "rule" For each of the number combinations, can you figure out the "rule" which alters the number?

What is a Function? Definition: A function is a rule Given X = { x1, x2, …} and Y = {y1, y2, …} Assign to each element of X a unique element of Y The set X is the domain the set of all possible x values The set Y is the range the set of all resulting y values

Notation y = f(x) can be thought of as .. “y is the image of the function f at x” or “y is the value of the function f at the point x” It is often read “y equals f of x” your instructor will often use the phrase “f at x …” this reflects the above

Alternate Definition A function is a set of ordered pairs: f(x) = { (x1, y1), (x2, y2), … } Where no two ordered pairs have the same first element Which of these is a function? { (3,4), (6,0), (9,4), (7,2)} {(1, 2), (3, 4), (5, 6), (1, 7)} {(8,-1), (9, -1), (10, -1), (105, -1)}

The -> is the STO> key Function Notation Normal notation: Functions can be defined/declared in your calculator: Note the use of functional notation f(3) to evaluate functions. The -> is the STO> key

Piecewise Functions Some functions may be defined differently for different portions of the domain. Your calculator can also define piecewise functions

Piecewise Functions Note the results when the function is graphed: Actually there should be a gap between 6 and the square root of 5 Which of the above values should apply for the function Try the F3, trace on the graph

Domain and Range Either the domain or range or both can be restricted due to the nature of the function Consider Determine the domain and range for each:

Domain and Range f(x) g(x) Domain: all real numbers BUT … not zero … why? Range: y < -2 or y > 2 g(x) Domain: x < -3 or x > -2 (why?) Range: y >= 0

Composition of Functions Basic Concept: value fed to first function result fed to second function end result taken from second function Oft used notation: y = g(f(x)) or

Composition of Functions Example – given : Try these f(4) = ? g(f(4)) = ? f(g(-2)) = ? Try defining the functions on your calculator and using the notation !!

Assignment Lesson 1.3A Page 31 Exercises: 3, 7, 11, 15, 21, 25, 27, 37, 43, 49

Defining a Graph The graph of a function is: the set of all points (x,y) which … satisfy the function y = f(x)

Some Graphs are NOT Functions Which are not functions? Use the “vertical line” test.

Intercepts We are often concerned with where the graph intersects the axes x-intercepts => when f(x) = y = 0 y-intercepts => when x = 0, f(0)

Reference Table of Functions Page 27, Text Linear Quadratics Cubic Absolute Root functions Reciprocals Trig functions

Symmetry of Graphs Symmetric with the y-axis f(-x) = f(x) for all x Called even functions

Symmetry of Graphs Symmetric about the origin f(-x) = -f(x) Called odd functions Note: There are many functions which are neither odd nor even

Transformation Shift a function up or down y – k = f(x) k > 0 => up k < 0 => down Shift function right or left y = f(x – h) h > 0 => right h < 0 => left

Try for the following function: Transformation Define the function on the home screen Enter different versions of f(x + h) f(x)+k a*f(x) f(-x) -f(x) Try for the following function:

Assignment Lesson 1.3B Pg 31 Exercises 51, 53, 57, 59, 63, 67, 69