Introduction To Functions

1. Introduction To Functions

2. The Function Machine

3. A Function In Action Put a large peperoni pizza into our function machine.

4. A Function In Action

5. A Function In Action What about an apple?

6. A Function In Action What will happen when we put a number into the function machine?

7. Function Terminology • A function is named with a single letter. We will call our function machine f. We can use any letter to designate a function i.e. h, b, t.

8. Function Terminology Lets call the thing being put into the machine (x). We would write f(x) = x/2 We say “f of x equals x over 2”

9. Function Terminology Relation Vs. Function Relation- Each number you put in is associated with each number you get out. Function-Every 'input' number is associated with exactly one'output' number. All functions are relations. Not all relations are functions.

10. Function Terminology The two definitions seem very similar, but taking a closer look we will identify the differences. Can you see the difference? Relation Function -2 2 4 22 1 3 7 12 3 9 21 64 -4 -3 -2 -1

11. Function Terminology Domain The domain of a function is the set of 'input' numbers for which the function is defined. In the function on the previous slide, the domain is {-2, 2, 4, 22}. Note: The set of input numbers are the values of x. All the values of x make up the domain.

12. Function Terminology Range The range of a function is the set of results to the equation for a given input. A true function only has one result for every domain. In the function on slide 11, the rangeis {-4, -3, -2, -1} Note: y is the variable associated with range. It can also be expressed as f (x). Also, y or f (x) is considered to be the output values of the function.

13. Function Terminology • f (x) = x/2 • Independent Variable – The variable x (input) is considered to be the independent variable because its value determines the value of other variables. • Dependent Variable – The variable y or f (x) (output) is considered to be the dependent variable because it’s value is determined by the value of the variable x.

14. Function Forms Combining all of our new ideas let’s take a look at how functions may appear. Using our function from slide 11, we can take the input and output values to form a set of ordered pairs. The x values (input/domain) are {-2, 2, 4, 22} The y or f (x) values (output/range) are {-4, -3, -2, -1} (x,y) =>{(-2,-3),(2, -4),(4, -2),(22, -1)}

15. Function Forms Using our function machine f We found that anything that was put into the machine was cut in half. We determined that the equation form was f (x) = x/2 More examples of equation form: g (x) = 3x + 4 t (x) = (-2/3)x – 16 p (x) = 10x

16. Function Forms Graphical Representation This is a graph of a function We can form this graph given: • The set of ordered pairs {(0,-4),(1, -2),(2, 0),(3, 2)} • The equation f (x) = 2x-4 • We can even use this graph to form both the equation and ordered pairs.

17. Real World Practice Now It’s your turn . . . Everyone must work together and respect each other at all times. See the handout for more detailed instruction.

18. Credit • Slide 2,3,4 – Function Machine: http://www.carlisleschools.org/webpages/wolfer/virtual.cfm • Slide 3,4 – Pizza: http://www.rivercitypizza.com/ • Slide 5 – Apple: http://fostertech.wordpress.com/2010/07/28/ios-4-iphone-configuration-utility/ • Slide 5 – Half Apple: http://www.istockphoto.com/stock-photo-7523903-red-apple-cut-in-half.php • Slide 6 – Number 100: http://www.minnesotafirsthome.com/buyers/100-financing-mn-first-time-buyers-only/ • Slide 6 – Number 50: http://www.radiowaves.co.uk/r/barto99 • Slide 16 – Graph: http://www.algebra.com/algebra/lessons/graphing/linear.epl