Differentiating functions and relations:

Telling a function and relation apart are crucial to identifying a function and being able to predict values beyond what you can see (extrapolate)

The main thing to look for is if any x value repeats even once, if it does, the pattern is just a relation of numbers and not a function where every output would have a unique input.

Lines on a graph:

When you have patterns graphed out, you can imagine a vertical line scanning through the graph and if at any point there are either two points or two segments of the line intersecting our imaginary scanner, that means an x value gets repeated at least once meaning it is not a function.

 

Table of values and Coordinates:

When working with a table of values or a list of coordinates, again all you have to look for is if an x value or input is repeated, if it does, it isn’t a function.

Notation and solving:

Writing a function has a few steps, first you need to know the rule of your pattern which in this example I’ll use 3x + 1, and the name of the function which is typically ƒ if only one is present, but can still be any other letter of the alphabet, there are two different notations of writing functions, mapping and function notation, mapping notation looks like y -> 3x + 1 and function notation looks like ƒ(x) = 3x+1.

after knowing your rule, solving for any other number comes down to inserting it as x and doing the rest of the math, if we were to input 5 into this function, the math would go 3 x 5 + 1 which would be 16.

In a case where you have a question like:
ƒ(x) = 3x + 1
f(x) = 25
you instead solve for what the x value would be which would go like this:
25 = 3x + 1, reciprocate the 1 and subtract from the 25 giving you 24, and divide both sides by 3, giving you 8 meaning that f(8) = 25 here.