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FUNCTIONS AND MODELS

FUNCTIONS AND MODELS . CHAPTER 1, SECTION 1 . Function Definitions. A function is a rule or correspondence that assigns to each element of one set exactly one element of a second set.

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FUNCTIONS AND MODELS

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  1. FUNCTIONS AND MODELS CHAPTER 1, SECTION 1

  2. Function Definitions A function is a rule or correspondence that assigns to each element of one set exactly one element of a second set. The set of elements in the first set is called the domain. If x is any element in the domain, then it is called the independent variable. The set of elements in the second set is called the range. If y is an output of the function from an input x, then y is called the dependent variable. Functions may be defined as a set of ordered pairs, a table, a graph, an equation or a verbal statement.

  3. Example: Find the domain. a) b) c) To make sure that the function results in a real number value, watch for two types of exceptions: Values from the domain that make the denominator 0 (zero). Values in the domain that make a negative value under an even root.

  4. For ordered pairs, if the independent variable values are all different, the table is a function. For equations, if the power on the dependent variable(y) is odd, then the equations defines the dependent variable(y) as a function of the independent variable(x). In the case of graphs, the Vertical Line Test is the way to determine if the graph represents a function. Vertical Line Test: a graph is a function if no vertical line intersects the graph in more than one point. Recognizing Functions Is the table a function? Does the equation define y as a function of x? Does the equation define y as a function of x?

  5. a) c) b)

  6. Function Notation We can use the function notation , read “y equals f of x” to indicate that the variable y is a function of the variable x. For specific values of x, f(x) represents the resulting y values especially when referring to a graph generated by the function. The point (a, f(a)) lies on the graph for any given a in the domain of the function. For example: represents a function and for

  7. Mathematical Models The process of transforming real-world information into a mathematical form that can be applied and interpreted is called modeling. For this reason, a mathematical model is a functional relationship that includes the function and a description of all variables and their units of measure.

  8. During the first two weeks in May, a man weighs himself daily(in pounds) and records the data. • Does the table define weight as a function of the day in May? • What is the domain? • What is the range? • During what day(s) did he weigh the most? • During what day(s) did he weigh the least? Example: Weight

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