Functions Algebra of Functions

Functions What are functions?

Functions What are functions? Possible answers Functions are relations in which no two ordered pairs have the same x- coordinate A set of rules for generating a specific output given an input

Linear Algebra, Abstract Algebra, Operator Algebra, Group Theory… Functions, or subsets of functions, may be treated as entities on which we perform operations, much like numbers Suggested Reading: Mario Livio, The Equation That Couldn’t Be Solved Deals not with functions, but with equations. This is related, since most functions we discussed are defined by equations.

Today Operations with Functions Addition and Subtraction Multiplication and Division Composition

Addition and Subtraction The domain of the sum or difference of functions is the intersection of the two domains That is, the function must exist for both f and g

Multiplication and Division For division only, the quotient function does not exist for any values of x for which the denominator is 0

Note on Domain Restrictions If the domain must be restricted in order for the function to exist, then that restriction must be noted! It is a trait of the function. Sometimes, a domain restriction is implied, but you should steer clear of that ambiguity

Composition of Functions Apply the internal function (f), then apply the external function (g) to that result The range of f must be contained in the domain of g If this is the case, then the domain of the composition is the domain of f

Practice Given the functions f and g, evaluate each of the following:

Activity, Exercise Thursday 4, 5, 7, 8, 9

Functions Compositions Domain and Range considerations Identity and Inverse

Today Compositions Identity and Inverse

Mapping Diagrams for Compositions (P.O.D.) All problems refer to the general function “g o f” Draw Mapping Diagrams for compositions in which The range for f is the same as the domain for g The range of f is a subset of the domain of g The domain for g is a subset of the range of f The range for f and the domain for g are disjoint Challenge: Can you come up with examples for f and g that correspond to each diagram?

The Point of that Exercise Mapping Diagrams An important tool in all algebras and other fields Domain/Range requirements (two ways to say it) The image under the first mapping must belong to the domain of the second mapping The range of the first function must be contained in the domain of the second function

Does a Composition Exist? Tools for testing Domain/Range tables The is one way to display the range and domain of each function Remember the goal: in order for a composition to exist, the range of the first function must be contained in the domain of the second function

Example Determine the implied domains and ranges, and create a domain/range table Does the composition exist? If so, determine the equation domainrange f g

Note on Existence You should have determined that the second composition does not exist because a logarithm of anything less than or equal to zero is undefined, BUT… The domain of the quadratic function can be restricted so that its range is all positive

Composition Review Do you know what a mapping diagram is? Do you know what a composition is? Do you know how to evaluate compositions? What has to be true about domains and ranges in order for a composition to exist? Do you know how to restrict a domain to allow a composition to exist?

Problem to show me that you know these things Page 154, 3x If one of the compositions does not exist, show how you could restrict the domain of the first function to allow it to exist

Activity, Exercise more problems 1, 2 3 (parts 5, 10, 13, 16) If the composition does not exist, can you restrict a function’s domain in order to make it exist? 5, 8, 12, 14, 16, 17, 21 Done last class? 4, 5, 8, 9, 14

Functions Inverse Functions

Today Problem of the Day Some homework review Review of Compositions Identity and Inverse Functions Domain and Range limitations Graphing Inverse functions by hand Graphing Inverse functions by GDC

Problem of the Day Show that the composition does not exist Restrict the domain of g to allow it to exist Give the equation for the composition Determine the composition’s inverse

Identities and Inverses: Review Addition What is the Identity for Addition? For a given number a, what is the additive inverse? Multiplication What is the Identity for Multiplication? For a given number a, what is the multiplicative inverse?

Identity and Inverse Identity Function Inverse Function Implies the “reverse” operations Name: “f inverse”

Domain and Range Limits If the inverse function of f (“f inverse” or “f -1 ”) exists, then: f must be a one-to-one function The domain of f is equal to the range of f -1 The range of f is equal to the domain of f -1

Obtaining the Equation for Inverse Functions Switch the independent and dependent variables Isolate the NEW dependent variable Note that ordered pairs are “reversed”

Using a Calculator To graph an inverse function, use the Draw menu: DrawInv [type the equation]

Graphing by Hand Graph the function f Graph the Identity function I(x) = x Treat I(x) as an axis of symmetry for the reflection of the function f Note: The point (a,b) becomes (b,a)

Graphing Practice Graph each function on the right Graph the inverse function by hand Check with your calculator Determine the inverse function algebraically

Functions vs. Relations Note that the functions should be one- to-one in order to have an inverse function Graph the inverse of y = x 2 Is it a function?

Exercise Don’t do these (save for next class)  3, 5 (a-d) Homework below: 6, 8 (left column on each) 10, 18 Draw a mapping diagram for a composition “g of f” such that the range of f and the domain of g intersect

(a, e, h) 18

Functions Transformations

Functions, So Far Graphing Exponential and Logarithmic Functions Operations with Functions Identity and Inverse Functions Compositions This week: Transformations

Activity Use your GDC to graph the functions on the right Describe (in words) how the functions on the right differ from the function y=x 2

Transformations Transformations of graphs are changes in location, shape or orientation The types of transformations are: Translations (horizontal and vertical) Dilations (horizontal and vertical) Reflections (about the x- and y-axes)

Horizontal Translations A horizontal translation moves the graph “a” units to the right

Vertical Translations A vertical translation moves the graph “b” units up

Combined Translations A horizontal translation moves the graph “a” units to the right A vertical translation moves the graph “b” units up

Vector Translations (Notation) Each point has coordinates (x,y), which can be written as a column vector (x on top, y on bottom) Vector translations can be described the same way

Vector Translations (cont.) “x-prime” and “y-prime” are the new x and y coordinates after the transformation (which, in this case, is a translation)

Labeling Graphs The MINIMUM for labeled graphs Title (the function) Label your axes Label axes intercepts Label the asymptote, vertex or any other important trait of the graph Use at least one other point to indicate the shape of your graph The more points you use, the easier it will be to fit a curve

Practice with Translations Exercise (a and c) 3c 4c 10d

Translations Homework Translations (6.1) 6, left column 8 (left column) 9 (a, b, d only) 10 (c only) Graph paper is required

Dilations Dilations can be in the form of a Stretching or Shrinking of the graph They may also be “in the y direction” or “in the x direction” This corresponds to “vertical” or “horizontal” Note on the textbook’s language (“from” axes)

Dilations in the y direction A dilation in the y direction multiplies all y- coordinates by p If IpI is greater than 1, this is stretching If IpI is less than 1, this is shrinking

Dilations in the x direction A dilation in the x direction multiplies all x- coordinates by (1/q). Note that “q” is the reciprocal of the coefficient If the coefficient is greater than 1, this is shrinking If the coefficient is less than 1, this is stretching

How to graph dilations Graph the original function For dilations along the y-axis Multiply each y-coordinate by p For dilations along the x-axis Multiply each x-coordinate by (1/q) Note that is x has a coefficient that is greater than 1, “q” is the reciprocal of that coefficient

Combined Transformations In general, follow the order of operations (PEMDAS), reversing operations and the order when IN the function That means that the transformations fall in this order (from “deal with them first”) Dilations along the x-axis Horizontal translations (this may go first) Dilations along the y-axis Vertical translations

Dilation/Translation Combinations Given the combined transformation, the form on the very bottom is preferred Challenge: Complete the sentence below: A horizontal dilation by a factor of 1/5 is equivalent to a vertical dilation by a factor of ___

Practice with Dilations Exercise 6.2 2b 3 6 Graph paper is required

Today Practice/Review of Translations and Dilations Homework Questions Reflections Odd and Even Functions

Reflections Reflections involve transferring the points on a graph across a line, or “reflecting a graph about a line” Today we will discuss reflections across the x- and y-axes Note: how does one draw the graph of an inverse function

Reflections about the x-axis A reflection about the x-axis involves changing the sign of all y-coordinates

Reflections about the y-axis A reflection about the y-axis involves changing the sign of all x-coordinates

Combined Transformations In general, follow the order of operations (PEMDAS), but reverse the order within the parentheses That means that the transformations fall in this order (from “deal with them first”) Horizontal translations Dilations along the x-axis Dilations along the y-axis Vertical translations Note: Reflections take effect with coefficients

Practice with Reflections Exercise (a and c) 3 (a, L*, o) Start with the parent function and build the graph through transformations. Check with your calculator. Challenge: In general, what graph would you obtain if you were to reflect y = f (x) about the line y = -x ?

This Week Next Class Complete the Transformation unit Start to review functions Thursday, there will be an assessment: Operations with functions Composite functions Inverse functions Transformations (Note: this is 5.4 through 6.3)

Test Functions Familiarity with different types, domain, range Operations with Functions Composite Functions Inverse Functions Defining and graphing them Transformations Translations, Dilations, Reflections

Geometers’ Sketchpad Activity Plot a parent function Choose values for a, b, c, and d (both positive and negative) Plot the transformations, one at a time, in the correct order, to get to the final function

Review Exercises Operations and Composites 5.4.1: 3*, 6, 7, 14 Inverse Functions 5.4.2: 1*, 3, 5*, 6*, 12 Transformations 6.3: 3 (left column), 4 (left column)

Revision Set A (page 233) 2, 4, 6, 7, 9a, 14, 15 On #7, could you combine the transformations from a, b and c? How about a, b, c, and d?