Unit 2

Function Sense

Phone a friend What is a Function

Objectives

• Determine the equation (symbolic representation)

• Determine the domain and range of a function

• Identify the independent and dependent variables of a function.

Input/Output

• There are two input variables that determine the cost (output) of filling up your car. What are they? Be specific.

The number of gallons pumped and the price per gallon of the gas are the two variables

Fill’er up

• Assume you need 12.6 gallons to fill up your car. Now one of the input variables will become a constant. The value of a constant will not vary throughout the problem. The cost of a fill-up is now dependent on only one variable, the price per gallon

Fill’er up• Complete the following table: for 12.6 gal tank

Fill’er up

Is the cost of a fill-up a function of the price per gallon? Explain.

Yes, for each value of input (price per gallon), there is one value of output (cost).

Write a verbal statement that describes how the cost of a fill-up is determined.

The cost is the price per gallon multiplied by 12.6

Fill’er• Let p represent the price of a gallon of gasoline pumped

(input) and c represent the cost of the fill-up (output). Translate the verbal statement into a symbolic statement (an equation) that expresses c in terms of p.

The cost is the price per gallon multiplied by 12.6

c = 12.6 p

Fill’er up

• The symbolic rule (equation) c=12.6p is an example of a second method of defining a function. Recall that the first method is numerical (tables and ordered pairs).

Function Notation

• The equation c = 12.6p may be written using the function notation by replacing c with f(p)

• f(p) = 12.6p

• Now if the price per gallon is $3.60, then the cost of a fill-up can be represented by f(3.60). To evaluate f(3.60, substitute 3.60 for p in f(p) = 12.6p as follows

• f(3.60) = 12.6(3.60) = 45.36

Fill’er up

• Using function notation, write the cost if the price is $2.85 per gallon and evaluate. Write the result as an ordered pair

f(2.85) = 12.6(2.85) = $35.91;(2.85, 35.91)

Real Numbers• What is a Real number?

• What are rational and irrational numbers?

Real Numbers

• Rational Number-is any number that can be expressed as the quotient of two integers .

• Irrational number-is a real number that cannot be expressed as a quotient of two integers

Domain and Range• Can any number be substituted for the input

variable p in the cost-of-fill-up function?

• Describe the values of p that make sense, and explain why they do.

Negative numbers do not make sense as input values. A value of 0 would mean that the gas is free. It would be unlikely to have a gallon of gas less than $1 or more than $5 ( well we hope).

Domain

• So what is a good viable definition for the domain of a function?

Domain- Is the collection of all possible values of the input or independent variable

Practical domain---Word problems

•Practical domain is the collection of replacement values of the input variable that makes practical sense in the context of the situation.

Domain

• Practical domain of the cost-of-fill-up function• The practical domain is the collection of real numbers

from 1 to 5 dollars

• The domain for the general function defined by c=12.6p, with no connection to the context• The domain is the set of real numbers, since the

expression 12.6p is defined for any real number replacement for the variable p.

Range

• So what is a good viable definition for the range of a function?

Range is the collection of all possible values of the output or dependent variable

Range

• The practical range corresponds to the practical domain

Range

• What is the practical range for the cost function defined by f(p) = 12.6p if the practical domain is 2 to 5?The practical range of this function is all real

number f(p) from 25.20 to 63

$25.20 is the cost at $2 per gallon$63 is the fill-up cost at $5 per gallon.

Range

• What is the range of this function if it has no connection to the context of the situation?

What is the range of this function if it has no connection to the context of the situation?

Domain and Range• Consider the following table that gives the

percentage of mothers in the workforce from 1999 to 2004 with children under the age of 6.

Independent variable= YearDependent variable- percentage

Domain - {1999,2000,2001,2002, 2003,2004}Range- {62.2,62.9,64.1,64.4,65.3}

Summary

• Independent variable is another name for the input variable of a function

• Dependent variable is another name for the output variable of a function

• The collection of all possible replacement values for the independent or input variable is called the domain of the function. The practical domain is the collection of replacement values of the input variable that makes practical sense in the contest of the situation.

Summary

• The collection of all possible replacement values for the dependent or output variable is called the range of the function. When a function describes a real situation or phenomenon, its range is often called the practical range of the function.

Summary

• When a function is represented by an equation, the function may also be written in function notation.

• y=2x + 3 as f(x)=2x+3