More advanced functions

Composition operation: means putting one function into another
Syntax:
f(x) = f[g(x)]
g(x) = g[f(x)]
The bigger letter is the function that is the external one
[ ], just means nothing and is just normal parentheses (), not something special
OR F(x) = f[z = g(x)], where z is just the variable for the bigger function (not important at all)

Identification of a graph that gets changed, like reflected over the x-axis is to simply take the  f(x)/y value and plug it into the new function and see which function gives you the right answer back

Variation:
Direct variation:

Inverse variation:

The Vertical-Line Test:
If you add a vertical line to a function anywhere, and it hits the function more than once, then it is NOT a function

Absolute value equations:
The way to solve an absolute value is to have two versions of the function
Ex.
|x-3|= 8
x-3 = 8 AND -(x-3) = 8
x = 11 and x = -5
ALSO
You can’t get a negative value of an absolute value equation
Ex.
|x-1| = -1, has no solution

The theory behind zeros:

c is just used as a representation of the x-value that creates the zero

Piecewise functions:
Smaller functions into one function, with their own sub-domains

Greatest Integer Function:
That weird door step function
Doesn’t have any decimal values only integers

How to decompose a composition of functions:
Just look at the function and find a possible g(x) and f(x)

Jointly variation Example:

And then you can find y, when you know x, z and now k too

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Invertible Functions:
A function that only has unique x and y values
Ex.
Quadratic functions are not invertible, but linear ones are