FUNctions!!!

This khan academy video was helpful for me to understand what makes a relation a function or not a function:

Math blog 1
FUNctions!
Tuesday, Jan 8:

Class started out with a couple verbal descriptions of functions:
“The time we arrive at Mt. Katahdin is a function of the time we leave Portland.”
“When we get to Acadia is a function of how fast we drive.”
“Our enjoyment of this hike is a function of the weather.”

Next, we had to match up a series of function descriptions to some graphs

We then looked at some examples of relations that are functions, and relations that are not functions. Based on those examples, we brainstormed what made a function, and this is what we came up with:
-A function exactly one value of y for each value of x
-If a vertical line goes through two or more points on the graph of a relation, than the relation is not a function.
-If it has a y is raised to a higher power than the power of 1, it will not be a function.

After we came up with these things, we went back to the examples of functions and non-function, looking at why each was which.
x=-9 is not a function because it is a vertical line, therefore there are infinite values of y for only one value of x, breaking the just one value of y for each value of x rule.

y=-9 is a function because it if you were to draw a vertical line through it at any place, it would not intersect more than once, because there is only one value of y though there are infinite values of x.

Sideways parabolas are not functions because if you put a vertical line through a function then it would intersect not once, but twice, meaning that there are two values of y for one value of x.


NOTATION:
Function notation: f(x)
-It reads “f of the x”
-NOT to be confused with algebraic notation; f is not a variable

After we talked a bit about function notation, we were given a function for which we were to find the f of x when x equaled different numbers:
Function: f(x)=3x2+2x

Find:
f(0)=0
f(1)=5
f(a)=3a2+2a
f(a+b)=3(a+b)2+2(a+b)
  ⇑
DON’T distribute here like you would with algebra

To wrap up class, we learned two definitions:
Domain: The set of all input values for a function
Range: The set of all output values for a function

And then found the domain and the range of a couple of the examples of functions:

x	f(x)
1	7
2	13
3	19
4	18

Domain: {x|1,2,3,4}
Range: {f(x)|7,13,18,19}
                  ⇑
This line means “such that”
This IS a function because it has just one value of y for each value of x

Wednesday, Jan 9:

Today we talked about absolute values and inequalities in relations.

Absolute value: The absolute value of a number is the number's distance from zero.

This is how absolute value looks:

|x|

Lets say that |x|=5

In that case, x could equal either 5 or -5, because those are the two numbers that are 5 away from zero. Absolute values are always positive because it is a distance, and a distance cannot be expressed in a negative number

|-3|=3

|2|=2

After we went over the meaning of absolute value, we started to look at absolute values in functions and relations:

The graph of y=|x|

This is a graph of the function y=|x|. The domain is all real numbers because it goes endlessly to both the left and the right. The ray going up to the right is for if x is positive, and the one going to left is for if x is negative, both are possible. We know it is a function because if you were to draw a vertical line through it, it would intersect only once. Its domain is all real numbers, and its range is y≥0.

Here is an example problem from class of absolute value relations:

{x,y:|y|=x and x≤3}

If you were to graph the integer points of this graph, it would look like this:

The domain would be {x|0,1,2,3} instead of just being all integers greater than or equal to zero because x is restricted by the inequality x≤3. It is not a function because if you were to draw a vertical line through it, it could go through 2 points.

Another khan academy video that might help explain absolute value inequalities. It has some expansions of ideas, especially towards the end:

Hope this helps!

-Isabel