1. Function Sense
Unit 2
Phone a friend
What is a Function

2. Objectives Determine the equation (symbolic representation)
Determine the domain and range of a function
Identify the independent and dependent variables of a function.

3. 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

4. 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

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

6. 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

7. 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

8. 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).

9. 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

10. 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)

11. Real Numbers What is a Real number?
What are rational and irrational numbers?

12. 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

13. 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).

14. 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

15. 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.

16. 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.

17. 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

18. Range
The practical range corresponds to the practical domain

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

20. 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?

21. 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= Year
Dependent variable- percentage
Domain - {1999,2000,2001,2002,
2003,2004}
Range- {62.2,62.9,64.1,64.4,65.3}

22. 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.

23. 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.

24. 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