Definitions: Sum, Difference, Product, and Quotient of Functions

Slides:



Advertisements
Similar presentations
Write an equation given the slope and a point
Advertisements

2.6 Combinations of Functions; Composite Functions Lets say you charge $5 for each lemonade, and it costs you $1 to produce each lemonade. How much profit.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
C OMBINATIONS OF F UNCTIONS ; C OMPOSITE F UNCTIONS.
Determine the domain and range of the following relations, and indicate whether it is a function or not. If not, explain why it is not. {(1, -4), (3, 6),
Inverse Functions. Objectives  Students will be able to find inverse functions and verify that two functions are inverse functions of each other.  Students.
Combinations of Functions; Composite Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
 Simplify the following. Section Sum: 2. Difference: 3. Product: 4. Quotient: 5. Composition:
1.7 Combination of Functions
Slide 1-1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Chapter 1 Graphs and Functions
Chapter 1 Functions and Their Graphs. 1.1 Rectangular Coordinates You will know how to plot points in the coordinate plane and use the Distance and Midpoint.
FUNCTIONS AND GRAPHS.
Graphs and Functions (Review) MATH 207. Distance Formula Example: Find distance between (-1,4) and (-4,-2). Answer: 6.71.
Combinations of Functions & Inverse Functions Obj: Be able to work with combinations/compositions of functions. Be able to find inverse functions. TS:
Combinations of Functions
Do Now Determine the open intervals over which the following function is increasing, decreasing, or constant. F(x) = | x + 1| + | x – 1| Determine whether.
SECTION 2.2 Absolute Value Functions. A BSOLUTE V ALUE There are a few ways to describe what is meant by the absolute value |x| of a real number x You.
2 Graphs and Functions © 2008 Pearson Addison-Wesley. All rights reserved Sections 2.5–2.8.
Functions and Models 1. New Functions from Old Functions 1.3.
Powerpoint Jeopardy Distance and Midpoint Formulas Determine the linear equation SymmetryDomain, Range, Max/Mins, Inc/Dec Miscellaneous
1.3 New functions from old functions: Transformations.
NEW FUNCTIONS FROM OLD New Functions from Old Functions In this section, we will learn: How to obtain new functions from old functions and how.
7.1 – Operations on Functions. OperationDefinition.
Chapter 2 Functions and Graphs Copyright © 2014, 2010, 2007 Pearson Education, Inc Combinations of Functions; Composite Functions.
Copyright © 2009 Pearson Education, Inc. CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra.
Review of 1.4 (Graphing) Compare the graph with.
Exponential and Logarithmic Functions
FUNCTIONS REVIEW PRE-CALCULUS UNIT 1 REVIEW. STANDARD 1: DESCRIBE SUBSETS OF REAL NUMBERS What are the categories? Where would these be placed? 7, 6,
Graphs and Graphing Utilities Origin (0, 0) Definitions The horizontal number line is the x-axis. The vertical number.
Operations of Functions Given two functions  and g, then for all values of x for which both  (x) and g (x) are defined, the functions  + g,
Ch1: Graphs y axis x axis Quadrant I (+, +)Quadrant II (-, +) Quadrant III (-, -)Quadrant IV (+, -) Origin (0, 0) (-6,-3) (5,-2) When distinct.
Chapter 1 vocabulary. Section 1.1 Vocabulary Exponential, logarithmic, Trigonometric, and inverse trigonometric function are known as Transcendental.
Review Chapter 1 Functions and Their Graphs. Lines in the Plane Section 1-1.
Copyright © 2004 Pearson Education, Inc. Chapter 2 Graphs and Functions.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Digital Lesson Algebra of Functions.
Quiz PowerPoint Review
Functions and Their Graphs RAFIZAH KECHIL, UiTM PULAU PINANG
Warm-up (10 min. – No Talking)
Basic Math Skills.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
College Algebra: Lesson 1
5.1 Combining Functions Perform arithmetic operations on functions
Functions Review.
CHAPTER 2: More on Functions
Mrs. Allouch JEOPARDY Unit 8.
Chapter 1 – Linear Relations and Functions
2.2 The Algebra of Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
The Algebra of Functions
2-6: Combinations of Functions
2.6 Operations on Functions
Combinations of Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 3 Graphs and Functions.
3.5 Operations on Functions
Warm Up Determine the domain of the function.
1.5 Combination of Functions
Chapter 1 Test Review.
Characteristics.
2.1 Functions.
CHAPTER 2: More on Functions
The Algebra of Functions
Characteristics.
Chapter 3 Graphs and Functions.
2-6: Combinations of Functions
f(x) g(x) x x (-8,5) (8,4) (8,3) (3,0) (-4,-1) (-7,-1) (3,-2) (0,-3)
The Algebra of Functions
Presentation transcript:

Definitions: Sum, Difference, Product, and Quotient of Functions Let f and g be two functions. The sum of f + g, the difference f – g, the product fg, and the quotient f /g are functions whose domains are the set of all real numbers common to the domains of f and g, defined as follows: Sum: (f + g)(x) = f (x) + g(x) Difference: (f – g)(x) = f (x) – g(x) Product: (f • g)(x) = f (x) • g(x) Quotient: (f / g)(x) = f (x)/g(x), provided g(x)  0 /

Example: Combinations of Functions Let f(x) = x2 – 3 and g(x)= 4x + 5. Find (f + g) (x) (f + g)(3) (f – g)(x) (f * g)(x) (f/g)(x) What is the domain of each combination?

Example: Combinations of Functions If f(x) = 2x – 1 and g(x) = x2 + x – 2, find: (f-g)(x) (fg)(x) (f/g)(x) What is the domain of each combination?

Review For Test 1 Equations of Lines Slope of a line, parallel & perpendicular Graph with intercepts, graph using slope-intercept method Formulae for distance and midpoint Graph circle with standard and general form (complete the square) Difference Quotient Piecewise-defined functions Domain & range of a function Intervals where the function increases, decreases, and/or is constant Be able to determine when a relation is a function – ordered pairs or graph Relative Max and Min Average Rate of change Even and odd functions, symmetry Transformations Algebra of functions, domain

The Composition of Functions The composition of the function f with g is denoted by f o g and is defined by the equation (f o g)(x) = f (g(x)). The domain of the composition function f o g is the set of all x such that x is in the domain of g and g(x) is in the domain of f.

Example: Forming Composite Functions Given f (x) = 3x – 4 and g(x) = x2 + 6, find: a. (f o g)(x) b. (g o f)(x) Solution We begin the composition of f with g. Since (f o g)(x) = f (g(x)), replace each occurrence of x in the equation for f by g(x). f (x) = 3x – 4 This is the given equation for f. (f o g)(x) = f (g(x)) = 3g(x) – 4 = 3(x2 + 6) – 4 = 3x2 + 18 – 4 = 3x2 + 14 Replace g(x) with x2 + 6. Use the distributive property. Simplify. Thus, (f o g)(x) = 3x2 + 14.

Example: Forming Composite Functions Given f (x) = 3x – 4 and g(x) = x2 + 6, find: a. (f o g)(x) b. (g o f)(x) Solution b. Next, (g o f )(x), the composition of g with f. Since (g o f )(x) = g(f (x)), replace each occurrence of x in the equation for g by f (x). g(x) = x2 + 6 This is the given equation for g. (g o f )(x) = g(f (x)) = (f (x))2 + 6 = (3x – 4)2 + 6 = 9x2 – 24x + 16 + 6 = 9x2 – 24x + 22 Replace f (x) with 3x – 4. Square the binomial, 3x – 4. Simplify. Thus, (g o f )(x) = 9x2 – 24x + 22. Notice that (f o g)(x) is not the same as (g o f )(x).