1. Relations and Functions

2. Relation – A relation between two sets is a way to match objects from one set with objects from another.
<<<< this is an example of a relation. Typically, you would express a relation with ordered pairs. So the relation from A to B would be {(1,2),(3,2),(5,7),(9,8)}.
A relation is any matchup, but a function is a type of relation. For a relation to be a function, no x-value can be related to two different y-values.

3. So, the relation we just saw is a function:
{(1,2),(3,2),(5,7),(9,8)}. Notice that No x-value is matched with more than one different y-value.
<<< this relation, however is not a function.
{(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)}
Notice that in this relation, 1 is related to 1, 2, and 3, and 2 is related to 2 and 3.
The x-value of 1 is related to more than one y-value, so it is not a function.

4. This is another way to map a relation by graphing
This is another way to map a relation by graphing. So the relation here from x to y would be {(0,-2),(0,1),(1,2),(2,1),(3,4)}.
The relation is not a function because the x-value 0 is mapped to more than one y-value.
You can also look at this relation and tell that it is not a function by using the “vertical line test”. You have all seen and done the vertical line test. If a vertical line touches more than one point, then the relation is not a function. Here, a vertical line on the y-axis would touch 2 points – hence, not a function.

6. Is it a function?
Drag the rectangle
to find out
Yes No No
Yes

7. Domain and Range
Domain – the set of values of the independent variable for which a function or relation is defined. Typically, this would be your x-values, but it doesn’t necessarily have to be. It is the set of “input numbers”.
Range – the set of values for the DEPENDENT variable. Typically y-values. The set of “output numbers”
So, for the relations that we saw earlier:
{(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)}
Domain: {1,2,3}
Range: {1,2,3}
{(1,2),(3,2),(5,7),(9,8)}
Domain: {1,3,5,9}
Range: {2,7,8}
Notice that repeated numbers don’t get repeated when you list the domain

8. Functions can also be defined by a definition. Here is an example:
f(x)=3x2+5
^^the easiest way to tell if this is actually a function would be to graph it. Your computer has an EXCELLENT graphing tool.
Click your search bar
Search for grapher.
This is the app that you want to use!

9. So, back to the function: f(x)=3x2+5
This is what your grapher should show you if you graph it. I had to zoom out a couple of times to see it. Notice that no vertical line touches more than two points, so this IS a function. Also notice that your grapher says y= instead of f(x)=
That’s because y and f(x) represent the same thing. It would be the same if you had g(x), h(x), z(x), or any other function “of x”.
You can do other fun things with the definition of a function also, like tell what the y-value (or output) is for any x-value (input).
Note: say I had a function h(y): then y would be the input, h the output…. Function t(r): then r would be the input, t the output.

10. For the same function we were just looking at: f(x)=3x2+5 let’s look at some actual values.
Find f(4): okay, notice that f(x), the x was replaced with a 4. So we need to do the same throughout the function. All we are doing is evaluating the function at that input.
f(4) = 3(4) = 3(16) = = 53
f(4) = 53.
When you input 4 into the equation, you get a 53 out. For the x value of 4, this function = 53……. Or you can look at it as a point. The point (4,53) would be on the graph of the function.
How about f(-1)
f(-1) = 3(-1)2+5 = 3(1)+5 = = 8
F(-1) = 8
The point (-1,8) is on the graph of this function.

11. Let’s look at an applied example of this
Let’s look at an applied example of this. In a circle, the area is a function of the radius. You’re used to seeing A=πr2. We could write this in function notation as A(r) = πr2.
So then, applying what we just saw, the area of a circle with radius 3 would be
A(3) = π*(3)2 = 9π
The area of a circle with radius 12 would be
A(12) = π*(12)2 = 144π
Another example: you are heating a pot of water. You figure out that the function that models this is h(t) = 5t+20 where t is the time in seconds, and h is the temperature in Celcius.
What is the heat after 5 seconds?
How long would it take for the water to boil? (water boils at 100˚C)
^^ answers on the next page

12. You are heating a pot of water
You are heating a pot of water. You figure out that the function that models this is
h(t) = 5t+20 where t is the time in seconds, and h is the temperature in Celcius.
What is the heat after 5 seconds?
h(5) = 5(5) = = 55˚C
How long would it take for the water to boil? (water boils at 100˚C)
For this question, you can guess and check OR you can set the equation equal to 100:
100 = 5t + 20
80 = 5t
16 seconds = t
^^ the pot was apparently heating very quickly, this is obviously not a realistic model.

13. One more example (IMPORTANT): you are shopping at a company that sells bundle packs of batteries. They sell you the first 8 batteries for $4, and each additional battery is $1 apiece.
Write a cost function for buying 8 or more batteries.
Then, find the cost of buying 16 batteries.
^^solution is on the next page

14. One more example (IMPORTANT): you are shopping at a company that sells bundle packs of batteries. They sell you the first 8 batteries for $4, and each additional battery is $0.75 apiece.
Write a cost function for buying 8 or more batteries.
C(x) = x
Then, find the cost of buying 16 batteries.
The first 8 are built into the function, because you have to buy 8 to get the deal. So you are buying 8 additional batteries.
Then C(8) = (8) = = $10

15. Back to domain and range!!
We already saw that given a set of coordinates, the x-values are the domain, and the y-values are the range.
But that is discrete, it is a set of specific numbers. What if we wanted to know the domain and range of a graph instead? Graphs are continuous.
You will have to use interval notation to describe the domain or range of a continuous graph (we used interval notation to describe number lines when doing inequalities).
Think back to inequalities like this.
5≤x 4 5 6
The interval notation would look like this:
[5, )
The domain and range of graphs will be represented in a similar manner.

16. Let’s look back at the graph we saw earlier: y = 3x2+5
Domain: any included x-value. You can’t tell very well from the picture, but the graph would keep going as far left and right as you could extend it if you could zoom out farther.
So the interval notation for the domain would be: ( , )
Range: any included y-value. Looking at the y-values of the graph, it looks like the graph begins at 5 and then goes up forever. So the interval notation for the range would be [5, )

17. One more: y = (square root (x-1)) +2
Domain: x-values. It looks like left to right, the graph begins at 1 and continues on forever, so the interval notation for the domain of this graph would be:
[1, )
Range: y-values. Looking up and down, the graph appears to start at 2 and to continue upwards forever, so the interval notation for the range would be: [2, )

18. Constant of variation: the number that shows the relation between two variables who are directly or inversely proportional to one another.
It is represented by the letter k.
Here is an example:
x y
5 10 16
4 8 20
^^^ in this example, y varies directly with x. The constant of variation is k = because to get any y-value, you multiply its corresponding x-value by 2.

19. Constant of variation is almost the same thing as slope.
Another example:
x y
2.5
8 4
4 2
10 5
16 8
In this example, the constant of variation is k = ½ because to get each y, you multiply its corresponding x by ½.
Constant of variation is almost the same thing as slope.
We could calculate the slope (m) of this set of data using the slope formula m = y - y
Just pick two points. I’ll use (4,2) and (8,4).
x -x
4-2
8-4 2 4 1 2
m = = = =

20. Find the constant of variation (slope) of this set of data:
x y
2 -6
I’ll choose the two points (0,0) and (10,-30)
7 -21
-1 3
0 0
m =
-30-0
10-0 =
-30 10 -3

21. Slope intercept form of a line: y = mx+b
Notice that y is BY ITSELF. Once you solve an equation for y (it may already be), you are in slope intercept form. m and b can be fractions, but don’t have to be. m is the variable that represents slope, and b represents the y-intercept.
So, for equation y = 4x-2, the slope is 4 and the y-intercept is -2.
So, to graph, plot the y-intercept at -2. Then since the slope is 4/1 (rise/run), go up 4 and over 1 from that point to get the next point on your line
Over 1
Up 4
y-intercept

22. EX. Graph the equation 5x+3y=6.
First, get it into slope-intercept form (get y by itself)
5x+3y = 6
3y = -5x +6
y = -5/3x <<<<< the slope (m) is -5/3. the y-intercept (b) is 2
So first, plot your y-intercept at 2. then, since your slope is -5/3, go down 5 and over 3 FROM THAT POINT.
y-intercept
Down 5
Over 3
(sorry the scale is weird)

23. Vertical and horizontal lines:
Vertical lines have no slope. We say that their slope is undefined because it can’t be measured. Any vertical line will have an equation of the form x = ###. The number is going to be where the line touches the x-axis.
For example, the line x = 4
Horizontal lines have a slope of zero (they are flat). Any horizontal line will have an equation in the form y = ####. The number is going to be where the line touches the y-axis.
For example, the line y = -3