Chapter 2 Notes.pptx - 2.4 Equations of Lines and Linear...

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2.4 Equations of Lines and Linear Models 2.3 The Slope of a Line 2.2 Linear Functions 2.1 Functions and Their Representations Chapter 2 - Linear Functions and Models
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2.4 Equations of Lines and Linear Models 2.3 The Slope of a Line 2.2 Linear Functions 2.1 Functions and Their Representations Chapter 2 - Linear Functions and Models
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2.1 Functions and Their Representations Function is a set of ordered pairs (x, y) where each x-value corresponds to exactly one y-value. Function notation y = f(x) is called function notation. The input is x and the output is y . The name of the function is f. Name of the function Output Input The y is the dependent variable , and x is the independent variable . The expression f(4) = 28 means that the result of the function is 28 when the input is 4. The letters , f, g, and h are commonly used to denote function names. )} 1 , 3 ( ), 3 , 2 ( ), 3 , 1 {( Ex. f ) ( x f y
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Four Ways to Represent Functions 2.1 Functions and Their Representations Table of Values X Input Y Output -5 -4 -2 -2 0 1 3 2 1. Graph 2. Numerical Representation Graphical Representation 4. Symbolic Representation 3 . Mapping Diagram -5 -2 0 3 -4 -2 1 2 Diagrammatic Representation The domain is the set of all x-values. The range is the set of all y-values. ) ( x f y } 2 , 1 , 2 , 4 { } 3 , 0 , 2 , 5 { R D
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2.1 Functions and Their Representations To evaluate a symbolic function, simply substitute the x in the function and solve for f(x). 13 ) 2 ( 7 6 ) 2 ( 7 ) 2 ( 3 ) 2 ( 2 where 7 3 ) ( a) f f f x x x f 2 . 0 5 . 2 5 . 0 2 5 . 0 5 . 0 ) 5 . 0 ( 5 . 0 where 2 ) ( b) f x x x x f 3 ) 10 ( 3 9 ) 10 ( 1 10 ) 10 ( 10 where 1 ) ( c) f f f x x x f
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2.1 Functions and Their Representations Using a real-world connection: Distance Traveled: A bicyclist is 4 miles from home, riding away from home at 8 mph. The function that represents the distance traveled is D(x) = 8x + 4 (symbolic representation) . a) Make a numerical representation that shows the bicyclist’s distance from home after 0, 1, 2, 3, and 4 hours. Hours (x) Distance (D) 0 4 1 12 2 20 3 28 4 36 b) Make a graphical representation that shows the bicyclist’s distance from home after 0, 1, 2, 3, and 4 hours.
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2.1 Functions and Their Representations The domain of a function is the set of all valid inputs. That is, all values which do no violate rules, such as division or square root. Find the domain of f . Since any value of x can be multiplied with -5 and added to 3, we say the domain is “all real numbers.” Since the denominator cannot equal 0, the domain can be any number except x=4. Domain is all numbers ≠ 4. Since the square root of negative numbers are not real numbers, x cannot be any negative number. Domain is all numbers ≥ 0. 3 5 ) ( a) x x f x x x f 4 2 ) ( b) x x f 4 ) ( c)
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2.1 Functions and Their Representations Find the Domain and Range from the Set of Data Domain: Range: All real numbers TRY THIS: Table of Values X Input Y Output 0 0 2 4 -2 4 3 9 -3 9 {y|y ≥ 0}
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2.1 Functions and Their Representations Is the set of ordered pairs a function?
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