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1 FUNCTIONS The values of one variable often depend on the values for another: The temperature at which water boils depends on elevation (the boiling point drops as you go up). The amount by which your savings will grow in a year depends on the interest rate offered by the bank. The area of a circle depends on the circle s radius. Dependent Variable Independent Variable dependent variable : the variable quantity that depends on the value of the another variable called the independent variable. function: a rule that assigns to each element in one set a unique element in another set. The sets may be sets of any kind and do not have to be the same. A function is like a machine that assigns a unique output to every allowable input. The inputs make up the domain of the function; the outputs make up the range. 1

2 When we define a function y f ( x) with a formula and the domain is not stated explicitly or restricted by context, the domain is assumed to be the largest set of x-values for which the formula gives real y-values the socalled natural domain. If we want to restrict the domain, we must say so. The domain of y x is understood to be the entire set of real numbers. We must write y x, x 0 if we want to restrict the function to positive values of x. The domains and ranges of many real-valued functions of a real variable are intervals or combinations of intervals. The intervals may be open, closed, or half-open, finite or infinite. The endpoints of an interval make up the interval s boundary and are called boundary points. The remaining points make up the interval s interior and are called interior points. Closed intervals contain their boundary points. Open intervals contain no boundary points. Every point of an open interval is an interior point of the interval. Name Graph Inequality Notation Interval Notation The set of all real numbers The set of numbers greater than a The set of numbers greater than a or equal to a The set of numbers less than b The set of numbers less than b or equal to b Open interval ab Closed interval ab Closed at a and open at b Open at a and closed at b Domain Range Domain Range

3 Recognize that the graph is reasonable. See all the important characteristics of the graph. Interpret those characteristics. Recognize grapher failure. Domain Range Domain Range EVEN FUNCTIONS AND ODD FUNCTIONS--SYMM The graphs of even and odd functions have important symmetry properties. For every x in the function s domain, a function is an Even function of x if Odd function of x if The graph is symmetric about the y-axis. The graph is symmetric about the origin. The graph of y x (an even function) is symmetric about the y-axis. 3 The graph of y x (an odd function) is symmetric about the origin. 3

4 Determine whether the function is even, odd, or neither. Try to answer without writing anything (except the answer). 1) y x 4 4 ) 3) y x 4) y x1 y x 1 While some functions are defined by single formulas, others are defined by applying different formulas to different parts of their domains. Piecewise functions let us make functions that do anything we want. A doctor s fee is based on the length of time. Up to 6 minutes costs $50. Over 6 to 15 minutes costs $80. Over 15 minutes costs $80 plus $5 per minute above 15 minutes Which we can write like this: $ 50 if t 6 f ( t) $ 80 if t 6 and t 15 $ 80 $ 5( t15 ) if t 15 a) If you were there for 1 minutes, what would the fee be? b) If you were there for 0 minutes, what would the fee be? Here is another piecewise function., if x 1 hx ( ) x, if x1 a) What is h(-1)? c) What is h(4)? b) What is h(1)? d) What is h(-5)? 4

5 Graphing Piecewise-Defined Functions Graph x, if x0 y f ( x) x, if 0 x 1 1, if x 1. Writing Formulas for Piecewise Functions Write a formula for the function y f ( x) whose graph consists of the two line segments in Figure f( x) Write a piecewise formula for the function. f( x) 5

6 The absolute value function y x is defined piecewise by the formula x x, if x0 x, if x 0. Domain Range, 0, The function is even, and symmetric about the y-axis. g( x) x h k vertex at ( hk, ) h 0 right; k 0 up h 0 left ; k 0 down Using Transformations (a) Draw the graph of the function. Then find its (b) domain and (c) range. 1) f ( x) x 1 ) f ( x) 3 x Domain Range Domain Range 6

7 FUNCTIONS The values of one variable often depend on the values for another: The temperature at which water boils depends on elevation (the boiling point drops as you go up). The amount by which your savings will grow in a year depends on the interest rate offered by the bank. The area of a circle depends on the circle s radius. Dependent Variable Boling temperature of water Amount of interest Area of a circle Independent Variable elevation Interest rate radius dependent variable : the variable quantity that depends on the value of the another variable called the independent variable. function: a rule that assigns to each element in one set a unique element in another set. The sets may be sets of any kind and do not have to be the same. A function is like a machine that assigns a unique output to every allowable input. The inputs make up the domain of the function; the outputs make up the range. a) b) 7

8 When we define a function y f ( x) with a formula and the domain is not stated explicitly or restricted by context, the domain is assumed to be the largest set of x-values for which the formula gives real y-values the socalled natural domain. If we want to restrict the domain, we must say so. The domain of y x is understood to be the entire set of real numbers. We must write y x, x 0 if we want to restrict the function to positive values of x. The domains and ranges of many real-valued functions of a real variable are intervals or combinations of intervals. The intervals may be open, closed, or half-open, finite or infinite. The endpoints of an interval make up the interval s boundary and are called boundary points. The remaining points make up the interval s interior and are called interior points. Closed intervals contain their boundary points. Open intervals contain no boundary points. Every point of an open interval is an interior point of the interval. Name Graph Inequality Notation Interval Notation The set of all real numbers The set of numbers greater than a The set of numbers greater than a or equal to a The set of numbers less than b The set of numbers less than b or equal to b Open interval ab Closed interval ab Closed at a and open at b Open at a and closed at b Domain Range, 0 0,, 0 0, Domain Range, 0, 0 8

9 Recognize that the graph is reasonable. See all the important characteristics of the graph. Interpret those characteristics. Recognize grapher failure. Domain Range, 0, Domain Range, 0, EVEN FUNCTIONS AND ODD FUNCTIONS--SYMM The graphs of even and odd functions have important symmetry properties. For every x in the function s domain, a function is an Even function of x if Odd function of x if The graph is symmetric about the y-axis. The graph is symmetric about the origin. The graph of y x (an even function) is symmetric about the y-axis. 3 The graph of y x (an odd function) is symmetric about the origin. 9

10 Determine whether the function is even, odd, or neither. Try to answer without writing anything (except the answer). 1) y x 4 4 ) 3) y x 4) y x1 y x x f ( x) x 4 4 x x f( x) 1 1 f( x) x x x 1 x 1 f( x) EVEN EVEN ODD NEITHER While some functions are defined by single formulas, others are defined by applying different formulas to different parts of their domains. Piecewise functions let us make functions that do anything we want. A doctor s fee is based on the length of time. Up to 6 minutes costs $50. Over 6 to 15 minutes costs $80. Over 15 minutes costs $80 plus $5 per minute above 15 minutes Which we can write like this: $ 50 if t 6 f ( t) $ 80 if t 6 and t 15 $ 80 $ 5( t15 ) if t 15 a) If you were there for 1 minutes, what would the fee be? b) If you were there for 0 minutes, what would the fee be? $80 $80 $5(0 15) $105 Here is another piecewise function., if x 1 hx ( ) x, if x1 a) What is h(-1)? c) What is h(4)? 4 b) What is h(1)? d) What is h(-5)? 10

11 Graphing Piecewise-Defined Functions Graph x, if x0 y f ( x) x, if 0 x 1 1, if x 1. Writing Formulas for Piecewise Functions Write a formula for the function y f ( x) whose graph consists of the two line segments in Figure x, 0 x1 f( x) x 1, 1 x. Write a piecewise formula for the function. x, 0 x f( x) 5 x, x

12 The absolute value function y x is defined piecewise by the formula x x, if x0 x, if x 0. Domain Range, 0, The function is even, and symmetric about the y-axis. g( x) x h k vertex at ( hk, ) h 0 right; k 0 up h 0 left ; k 0 down Using Transformations (b) Draw the graph of the function. Then find its (b) domain and (c) range. 1) f ( x) x 1 ) f ( x) 3 x Domain Range, 1, Domain Range,, 1

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