4.2 Relations and Functions
4.2 Investigation – What is a relation? What is a Function?
Purpose: To determine what a relation is and to determine the characteristics of a special relation called a function.
While both of the situations that follow are relations, only one is a function. Explore the following to determine the differences and the characteristics needed to qualify a relation as a function.
1)a) Jack earns $7.50 per hour while working. Give an equation to describe the relation between Jack’s earnings (y) and hours (x). Create a table of values for the relation for 0-5 hours.Graph the relation.
b)The following table describes how much Jill has earned while working at various times on various jobs. Graph the data.
x – hrsy - $ earned
00
16
110
215
315
320
424
530
550
540
2)a) One of the graphs in #1 illustrates a function while one is not. Discuss how they compare. (Hint: focus on the relationship between the x coordinates and the y coordinates in each).
b) True or false? Jack’s earnings depend on the hours worked.
Do Jill’s earnings depend on the hours worked.
c)Is it possible to predict how much Jack will make in 6 hrs? Why or why not? Is it possible to predict how much Jill will make in 6 hrs? Why or why not?
Graph/situation is a relation while ___ illustrates a special relation called a function.
Relation -
Function-
Tell whether each relation is a function or non function
3. a) b) c) d)
4. a) x yb) x yc) x y
0 0 0 0 0 5
1 1 1 5 1 5
1 2 2 10 2 5
2 2 3 15 3 5
3 3
5. a) (1,4) (2,5) (3,6) (4,7) b) (1,1) (1,2) (1,3) c) (1,1) (2,1) (3,1) (4,1)
6.a) y = 6x + 2 b) y = + c) y = x
7. Boys Girlfriends
Bo Beth
Luke Louise
Jake Jane
Function Notation
As we know we often use algebra to describe things. If it is used to describe a function we can use function notation. Recall that the equation used to describe Jack’s earnings in the previous activity was y = 7.50x Since that is a function we could use function notation and write the function f(x) = 7.50 x The y was replaced with f(x) since, in a function, y is function of x.
Example – snowing 2cm per hour
Equation y = 2x
Since that would be a function
-one y value for each x and y (snow) depends on x (time)
it could be written f(x) = 2x
If asked to find the amount of snow after 12 hours it would look as follows:
f(x) = 2x
f(12) = 2(12)
f(12) = 24
A nice feature of function notation is that if allows you to see the value of x (independent variable) and the value of the function (y-dependent variable) together. Example – at 12 hours there were 24cm of snow.
Answers to selected problems
2. a) On Jack’s graph each x coordinate has only 1 y coordinate.
On Jill’s some of the x values are partnered with more than 1 y value.
b) True, Jack’s earnings do depend on the hours worked.
False, Jill’s earnings do not depend on her hours worked.
c) Yes it is possible to determine how much Jack would make in 6 hrs since there is a pattern since his earnings depend on the hours worked.
No it is not possible to determine how much Jill would make in 6 hrs since her earnings do not depend on the hours worked.
Graph/situation B is a relation while A illustrates a special relation called a function.
Relation – any set of ordered pairs
Function - a unique relation in which the following characteristics are true:
-each x value is paired with only one y value
-y depends on x (or y is a function of x)
-predictions are possible
3. a) f b) n.f. c) f d) n.f. 4. a) n.f. b) f c) f
5. a) f b) n.f. c) f 6. a) f b) n.f. c) f
7. n.f.