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College Algebra Chapter 3 Operations on Functions and Analyzing Graphs College Algebra Chapter 3.1 The algebra and composition of functions Sums and Differences of Functions For functions f and g with domains of P and Q respectively, the sum and difference of f and g are defined by: f g x f x g x f g x f x g x College Algebra Chapter 3.1 The algebra and composition of functions Sums and Differences of Functions f x x 5 x find f g x ? 2 g x 2 x 10 f g x ? College Algebra Chapter 3.1 The algebra and composition of functions Sums and Differences of Functions f x x 5 x 2 g x 2x 10 if hx f g x what would h3 ? College Algebra Chapter 3.1 The algebra and composition of functions Products and Quotients of Functions f g x f x g x f f x x g x g g x 0 College Algebra Chapter 3.1 The algebra and composition of functions Products and Quotients of Functions f x x 2 g x x 3 find hx f g x evaluate h2 and h5 College Algebra Chapter 3.1 The algebra and composition of functions Products and Quotients of Functions f x x 2 4 x 12 g x x 2 7 x 6 f find h x x g state dom ain College Algebra Chapter 3.1 The algebra and composition of functions Products and Quotients of Functions f x x 2 9 g x x 1 find H x f g x f find hx x g state dom ain College Algebra Chapter 3.1 The algebra and composition of functions Composition of Functions f x x 2 g x x 2 hx f g x f g x College Algebra Chapter 3.1 The algebra and composition of functions Composition of Functions College Algebra Chapter 3.1 The algebra and composition of functions Composition of Functions f x x 2 x 3 g x x 3 2 find f g x f g x x 3 2x 3 3 2 College Algebra Chapter 3.1 The algebra and composition of functions Function Decomposition h x x 4 h x f g x g x ? f x ? College Algebra Chapter 3.1 The algebra and composition of functions Function Decomposition h x x 1 2 f g x f g x f x ? x 2 g x ? x 1 College Algebra Chapter 3.1 The algebra and composition of functions Homework pg 256 1-77 College Algebra Chapter 3.2 one to one and inverse functions Relations Functions One to One Function If a horizontal line intersects a graph at only one point, the function is one to one College Algebra Chapter 3.2 one to one and inverse functions Functions 1 to 1 functions College Algebra Chapter 3.2 one to one and inverse functions College Algebra Chapter 3.2 one to one and inverse functions Inverse functions An inverse function is denoted by 1 This does not mean f x If given coordinates (x,y) the inverse would have coordinates (y,x) (3,4) (-2,8) (-7,10) 1 f x College Algebra Chapter 3.2 one to one and inverse functions Inverse functions An inverse must undo operations taking place in the original equation College Algebra Chapter 3.2 one to one and inverse functions Inverse functions How to find an inverse Algebraically 3 f x x 4 3 y x 4 3 x y 4 3 x4 y x4 y 3 f 1 x 3 Use y instead of f(x) Interchange x and y Solve for y The result is the inverse x4 College Algebra Chapter 3.2 one to one and inverse functions Inverse functions f f x x 1 College Algebra Chapter 3.2 one to one and inverse functions Inverse functions f x 2 x 1 5 College Algebra Chapter 3.2 one to one and inverse functions Inverse functions f x 2 x 14 2 College Algebra Chapter 3.2 one to one and inverse functions Homework pg 268 1-96 College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Vertical shift or vertical translation Given any function whose graph is determined by y and k>0, 1. The graph of y f x k is the graph of f x shifted upward k units. 2. The graph of y f x k is the graph of f x shifted downward k units. f x The amount of shift is equal to the constant added to the function College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Vertical shift or vertical translation yx 2 College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Vertical shift or vertical translation y x 4 2 College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Vertical shift or vertical translation y x 2 2 College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Vertical shift or vertical translation f x x f x x 1 f x x 3 College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Horizontal shift or horizontal translation Given any function whose graph is determined by y and h>0, 1. The graph of y f x h is the graph of f x shifted to the left h units. 2. The graph of y f x h is the graph of f x shifted to the right h units. -Happens when the input values are affected -Direction of shift is opposite the sign f x College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Horizontal shift or horizontal translation y1 x 2 f x x y2 x 2 2 g x x 3 College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Horizontal shift or horizontal translation Graph f x x College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Horizontal shift or horizontal translation Graph f x x 4 College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical and Horizontal Shifts Horizontal shift or horizontal translation Graph f x x 2 College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical Reflection over x-axis y1 f x f x x 2 x 2 y2 f x g x x 2 x 2 College Algebra Chapter 3.3 Toolbox functions and Transformation Vertical Reflection over x-axis 1 f x x 2 3 2 1 f x x 2 3 2 College Algebra Chapter 3.3 Toolbox functions and Transformation Horizontal Reflections over y-axis y1 f x f x x 8 3 y2 f x g x x 8 3 College Algebra Chapter 3.3 Toolbox functions and Transformation Horizontal Reflections over y-axis f x x 8 3 g x x 8 3 College Algebra Chapter 3.3 Toolbox functions and Transformation y1 f x f x x 2 y2 a f x 1 2 g x x 3 College Algebra Chapter 3.3 Toolbox functions and Transformation Ways to graph transformations 1) Using a table of values 2) Applying transformations to a parent graph a. Apply stretch or compression b. Reflect result c. Apply horizontal or vertical shifts usually applied to a few characteristic points College Algebra Chapter 3.3 Toolbox functions and Transformation f x x 3 g x 4 x 3 College Algebra Chapter 3.3 Toolbox functions and Transformation Homework pg 283 1-86 College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form y ax h k 2 Horizontal shift is h units, vertical shift is k units To put a quadratic equation in shifted form can be done by completing the square College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square y 2 x 8x 3 2 y 2x 4x ____ 3 y 2x 4 x 4 4 3 y 2 x 8x ____ 3 Group variable terms 2 Factor our “a” 2 2 y 2 x 2 4 3 2 y 2x 2 5 2 Add and subtract then regroup 1 2 linear coefficient Factor trinomial Distribute and simplify 2 College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square f x x 4 x 5 2 f x x 2 9 2 g x x 5 x 2 2 2 5 17 g x x 2 4 College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square f x 3x 5x 1 2 1 2 g x x 5 x 7 2 College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square Go back 3 pages to find zero’s of each function Set equation equal to zero and then solve for x College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square y 2 x 8x 3 2 y 2x 4x ____ 3 y 2x 4 x 4 4 3 y 2 x 8x ____ 3 Group variable terms 2 Factor our “a” 2 2 y 2 x 2 4 3 2 y 2x 2 5 2 Add and subtract then regroup 1 2 linear coefficient Factor trinomial Distribute and simplify 2 College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square y 2x 2 5 2 0 2x 2 5 2 5 2x 2 5 2 x 2 2 5 x2 2 5 2 x 2 2 College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square f x x 4 x 5 2 f x x 2 9 2 x 5 or 1 g x x 5 x 2 2 2 5 17 g x x 2 4 College Algebra Chapter 3.4 Graphing General Quadratic Functions Shifted Form/Vertex Form Completing the square f x 3x 5x 1 2 1 2 g x x 5 x 7 2 College Algebra Chapter 3.4 Graphing General Quadratic Functions f x ax2 bx c Standard form for a quadratic function has a vertex at b b h, k , f 2a 2a College Algebra Chapter 3.4 Graphing General Quadratic Functions Homework pg 295 1-60 College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions 1 Reciprocal Functions x Reciprocal Quadratic Functions 1 x2 Asymptotes are not part of the graph, but can act as guides when graphing Asymptotes appear as dashed lines guiding the branches of the graph College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions Direction/Approach Notation As x becomes an infinitely large negative number, y becomes a very small negative number x 0 , y x , y 0 x , y 0 x 0 , y College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions Horizontal and Vertical asymptotes a F x k xh a Gx k 2 x h The line y=k is a horizontal asymptote if, as x increases or decreases without bound, y approaches k The line x=h is a vertical asymptote if, as x approaches h, |y| increases or decreases without bound College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions Horizontal and vertical shifts of rational functions First apply them to the asymptotes, then calculate the x- and y-intercepts as usual College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions 1 g x 2 x To find x intercept; solve 1 0 2 x 1 x 2 College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions 1 g x x2 College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions 1 g x 1 x2 To find y-intercept; solve 1 g 0 1 02 To find x-intercept; solve 0 1 1 x2 College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions Homework pg 307 1-56 College Algebra Chapter 3.6 Direct and inverse Variation College Algebra Chapter 3.6 Direct and inverse Variation College Algebra Chapter 3.6 Direct and inverse Variation College Algebra Chapter 3.6 Direct and inverse Variation Homework pg 321 1-58 College Algebra Chapter 3.7 Piecewise – Defined Functions Piecewise-Defined Functions College Algebra Chapter 3.7 Piecewise – Defined Functions Piecewise-Defined Functions What is the piecewise function? x 2 6 x 3 0 x 6 f x x3 3 College Algebra Chapter 3.7 Piecewise – Defined Functions Piecewise-Defined Functions College Algebra Chapter 3.7 Piecewise – Defined Functions Piecewise-Defined Functions College Algebra Chapter 3.7 Piecewise – Defined Functions Homework pg 335 1-42 College Algebra Chapter 3 Review •Composition of functions •Inverse function •One-to-one function •Transformation •Translation •Reflection •Quadratic •Absolute value •Linear •Reciprocal •Reciprocal quadratic function •Piecewise-defined functions •Effective domain College Algebra Chapter 3 Review Composition of functions f x x 2 5 f g x ? f g x ? f g x ? f 1 x ? f g x ? g x 5 x Domain and Range? College Algebra Chapter 3 Review Toolbox Functions Know them and their graphs College Algebra Chapter 3 Review f x 2x 3 3 3 2 g x 3 2 x 2 College Algebra Chapter 3 Review Variation The weight of an object on the moon varies directly with the weight of the object on Earth. A 96-kg object on Earth would weigh only 16 kg on the moon. How much would a 250-kg astronaut weigh on the moon? College Algebra Chapter 3 Review Piece-Wise Defined Functions .5 x 1 x 3 hx x 5 3 x 5 3 x 5 x 5 .5 x 1 hx 2 x 4x 1 x0 0 x5 College Algebra Chapter 3 Review