Algebra 1/Unit 4: Functions

Functions
This lesson should bring you back to primary, where you learned how to point a relation on a rectangular coordinate system. What's a relation? We should get right into that.

A relation is any set of ordered pairs. So when your teacher asks the class: Where is (5, 3) on the rectangular coordinate system? The  in the question is the relation. When we start examining the relation, we get into what is known as the domain and the range.

The domain is the x-coordinate point of the relation. So, when we have the relation (6, 2). We first look for the x-coordinate point, which is 6. After finding 6 on the map, we need to look for the range, which is the y-coordinate point in the relation. In (6, 2), the range is 2.

If we have a relation of coordinates, such as (6, 7), (8, 2), (1, 5), (9, 5), (2, 4), (7, 1), and (3, 8), you should tell the domains and ranges of these coordinate points. If you don't, don't worry, I got your back. The domains are: 6, 8, 1, 9, 2, 7, and 3, while the ranges are 7, 2, 5, 5, 4, 1, 8. As you can tell after this quick example, finding the domains and ranges of relations are quite easy and simple.

Sample problems for finding domains and ranges of relations
Moving on, we use the domains and ranges to find out if a relation is functional. To find out about whether an equation is a function or not, we will move on to (section 1).
 * (9, 3), (5, 6), (2, 3), (3, 2), (3, 8)
 * (4, 5), (2, 9), (7, 6), (6, 7), (3, 10)
 * (7, 8), (2, 3), (3, 4), (1, 9), (4, 19)
 * (5, 2), (1, 9), (1, 8), (1, 3), (3, 7)
 * (9, 3), (2, 4), (6, 4), (7, 0), (1, 2)
 * (8, 1), (2, 5), (5, 2), (6, 2), (-3, 5)

How to find out if a relation is a function?
Let us define what a function is. What is known as a function is a relation that does not have the same x-coordinate point.

An example of a function is y= 9x + 96 - 40. Why? Because whenever we add in an input for "x", we will get a different "y" output. So, if we put in 0 for x, we should get "56" for the y-output. When we add in 1 for x, we should get 65 as the y-coordinate. Same principle for 2 (you should get 74).

Now that we have several numbers, we can make a graph that represents the relationship between the x-input and the y-output. Just like so:

Now that we have a relation, let us examine, pretending that we don't know that this is already a function, this relation to see if this is a function or not. Are there any x-coordinates that are the same value? No, we don't. We have 0, 1, and 2. These three numbers are totally different numbers. On another side note (no need to worry about range, range is useless when it comes to finding out if a relation is a function or not), we have totally different y-outputs. Our different y-outputs are 56, 65, and 74. So now seeing that this relation has different domains, we can now declare that this relation is a function !!

What is Function Notation?
Symbol: $$f(x)$$

$$f(x)$$ does not mean "f times x". But $$f(x)$$ is simply $$y$$. Most of the time, we don't use the y, and we use function notation. So, when we have $$y=2x + 7$$. We advance, and replace the "y" with $$f(x)$$. So, it is $$f(x)$$=2x + 7.

An example of "Function Notation" is $$f(x)$$ = 0.53x - 17.03 (x = height of man in inches). This equation, $$f(x)$$ = 0.53x - 17.03, represents the approximate length of a man's femur (thigh bone). Say, if a man is 70in (inches), the equation would turn into f(70) = 0.53(70) - 17.03. Why? Because we have to replace all the "x"'s with "70" (the number that is "piecing a bit of the puzzle". After adding in the "70"s for the "x"s, we can move onto figuring out the solution to this problem using our Order of Operations skills). Now, after getting this clear/covered up, let's solve the problem:


 * 1) $$f(x)$$ = 0.53x - 17.03
 * 2) $$f(70)$$ = 0.53(70) - 17.03
 * 3) $$f(70)$$ = 37.1 - 17.03
 * 4) $$f(70)$$ = 20.07

Amazing! We, thanks to the wonders of math, have found out that the approximate length of a 70 inch man's femur would be about 20.07 inches. Pretty easy if you know your arithmetics!

Sample problems for finding function notation
 {$$f(x) = 8x - 45$$ $$f(x) =$${ 8x-45 (i)|8x - 45 (i) _22 }
 * type="{}"}

{$$f(x) = 93 + 22 - 5x + 9x$$ $$f(x) =$${ 4x+115 (i)|4x + 115 (i) _22 }
 * type="{}"}

{$$f(x) = 5x - 68$$ $$f(x) =$${ 5x-68 (i)|5x - 68 (i) _22 }
 * type="{}"}

{$$f(x) = \tfrac{5}{3}x + 22 (88 - 10.02)$$ $$f(x) =$${ 5/3x+42889/25 (i)|5/3x+42,889/25 (i)|5/3x + 42889/25 (i)|5/3x + 42,889/25 (i) _22 }
 * type="{}"}

{$$f(x) = \tfrac{1}{2}x + 71 (98 - 23.209)$$ $$f(x) =$${ 1/2x+5310161/1000 (i)|1/2x+5,310,161/1,000 (i)|1/2x + 5310161/1000 (i)|1/2x + 5,310,161/1,000 (i) _22 }
 * type="{}"}

How do I solve function notation using a chart?
This section will help you to solve function notation using a chart. Here, we have an example chart (which is colored in the hopes you are more interested):

Now, we will throw a question at you (not Jackie Robinson-throw, but just to give out): What is x of $$f(x)$$ = 2?

This question is, thankfully, very easy. All you have to do is closely inspect the graph. Find the $$f(x)$$ section, and find the number 2 in here. And find the x-input of "2". The number we get is "3". 3 is the $$f(x)$$ of 2! Why? If we plug in 3 to the $$f(x)$$ problem, we will get 2 (as you can see, it is recorded in the graph). Thus, 3 is the $$f(x)$$ of 2.

Quiz
 {$$f(x) = 8x - 45$$ { -37_15 }
 * type="{}"}

{$$f(x) = 93 + 22 - 5x + 9x$$ { 123_15 }
 * type="{}"}

{$$f(x) = 5x - 68$$ { -53_15 }
 * type="{}"}

{$$f(x) = \tfrac{5}{3}x + 22 (88 - 10.02)$$ { 129167/75|129,167/75|1722 17/75|1,722 17/75_15 }
 * type="{}"}

{$$f(x) = \tfrac{1}{2}x + 71 (98 - 23.209)$$ { 5312.661|5,312.661|5312661/1000|5,312,661/1,000|5312 661/1000|5,312 661/1,000_15 }
 * type="{}"}