Properties of a function that applies for all of the functions:
Transformation: By organizing the function into y=a f(b(x-h))+k, we can apply transformation such that a is vertical compression/expansion, b is horizontal compression/expansion, h is horizontal translation and k is vertical translation. Negative signs can be applied to a and b value so that it there can be reflection on x and y axis respectively.
Composition of functions: Given two functions f(x) and g(x), the composite function f [g (x)] is such that the domain of f(x) includes the range of g(x).
Inverse of a function: f(x) can be a function, however f^-1(x) would be the inverse of the function, such that for each x and y value in f(x), the x value will equal to the y value in the inverse function and the y value will equal to the x value of the inverse function. To have a inverse function, the function has to be one to one, however you can restrict the domain to find the inverse of it.
X and Y intercept: The x and y intercepts are coordinate values on the graph that is on x axis or y axis of the cartesian plane. The x coordinate of y intercept is 0. The y coordinate of x intercept is 0.
Domain and Ranges: The domain is the set value that can be used as the x value of the function. The range is the set value that can be used as the y value of the function.
Domain and ranges can also be described using [ ]and () brackets, such that when the value is included we can use bracket, when it is not included we can us () brackets. Values of a function are not included when there is a open circle
Properties of function that applies for some of the functions:
Asymptotes:The concept of asymptote is such that the x or y value of the graph will be infinitely close to the x or y asymptote line, however it does not reach it.
Holes: If the numerator and denomenator of the function has common factor, then there is a hole in the function, which is serves as a "point of discontinuity". It can be found through finding the factored component of x and equate it to 0 to find the x value of hole. The y value of the hole can be found through plugging the x value back into the simplified version of the function.