LEI - Functions (Lesson)
Functions
A relation consists of a set of ordered pairs (x, y); a relation can also be called a mapping. The x-values are the domain and the y-values are the range.
In the relation above, the domain is {1, 3, 5, 9} and the range is {2, 7, 8}. The range is only the values that have been "used" by the domain. We say the domain is the independent variables and are usually the x-values (what you input) and the range is the dependent variables and are usually the y-values (the output).
A relation can be mapped onto a graph by plotting each of the ordered pairs. This graph to the right maps the relation above.
A FUNCTION IS A RELATION IN WHICH EACH X - VALUE MAPS TO EXACTLY ONE Y - VALUE
In order for a relation to be a function, each x-value can only be associated with one y-value. It is OK if multiple x-values map to the same y-values!
| Function | Not a Function |
|---|---|
| {(3, 2) (5, 2) (1, 4) (6,3)} | {(3, 2) (3, 5) (1, 7) (6, 6)} |
Is it a function or not a function? Practice
- {(3, -1) (5, -1) (7, -1) (9, -1)}
- {(2, 3) (3, -1) (5, 6) (-2, 4)}
- {(9, 1) (-8, 2) (3, 6) (10, 1)}
- {(3, 5) (3, 2) (3, 1) (3, 7)}
- {(2, -1) (5, -1) (5, 6) (-1, 2)}
- {(-7, 8) (8, 9) (9, -7) (-7, 10)
TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.
Let's look at the graphical representation of relations and functions:
Notice that in a function, none of the points are on the same vertical line, but in the relation that is NOT a function two of the points are on the same vertical line.
Watch this video to get a better idea of how the Vertical Line Test works!
We use functions to tell us about relationships between values. For instance, let's say your cell phone plan charges you $0.15 per MB of data used. So we can write a function for the cost (C) in terms of the amount of data used. We would say C(m) = 0.15m.
Use the rule to complete the table for the given domain values. Write the result as an ordered pair.
|
Independent Variable: m MB's Used |
Dependent Variable: C(m) = 0.15m | (m, C(m)) |
| 1 | C(1) = 0.15(1) = 0.15 | (1, 0.15) |
| 10 | C(1) = 0.15(10) = 1.50 | (10, 1.50) |
| 25 | C(1) = 0.15(25) = 3.75 | (25, 3.75) |
| 100 | C(1) = 0.15(100) = 15 | (100, 15) |
| 180 | C(1) = 0.15(180) = 27 | (180, 27) |
A common misconception might be to think the domain for this function is {1, 10, 25, 100, 180} however, it is not! We know that you could use any amount of MB's of data. So we must account for those continuous values, not just the values we put in the table. The domain for this function would be: m > 0.
We know the amount of data used must be greater than 0, because you can't use a negative amount of data. But after that there are no restrictions on what the input could be!
Watch this video for a few more examples of how functions work:
Input Output Practice
- You bought a plant that is 5 inches tall and you know that the plant will grow at a rate of 2 inches per week. Write a function for the height, h, of the plant after a certain number of weeks, w.
- You are selling brownies at the bake sale for $0.75 each. Write a function for the revenue, R, you've earned based on the number of brownies you've sold, b.
- Evaluate the function f(x ) = 7x - 3 for each value below: f(3), f(-2), f(0), f(-5)
TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.
IMAGES CREATED BY GAVS