Functions

Asymptote

• A vertical asymptote occurs when the denominator is 0
• When there are roots, it cannot be negative. This leads to the graph stopping at that point.
• If a graph of a function approaches but does not reach a horizontal line, it has a horizontal asymptote

Types of relations

• Refers to the number of values each x or y coordinate has; e.g. a linear graph is one to one as there is only one place on the line which has a certain x value, and y value.
• Only one-to-one and many-to-one are functions

One-to-one

• There is only one point which has y = -1 (e.g. -1,-1)
• Example:
  

Many-to-one

• There are two or more places which have the same y value, but has unique x values for every part of the line
• Example:
  

One-to-many

• Has unique y values for many possible x values
• Example:
  

Many-to-many

• Has more than one place with the same x and y value
• Example:
  

Vertical line test

• To test if you have a function
• Any vertical line crosses the line in only one place (for functions)

Horizontal line test

• To test if a function is one-to-one or many-to-one
• Only crosses the line once if a line is one-to-one

Domain and range of a function

• Domain = the set of inputs (x values)
• Range = the set of outputs (y values)

Graph sketching

Key features

• Y intercepts
• X intercepts
• Roots
• Zeros
• Vertices

Composite functions

• (f ∘ g)(x) = plug function g inside of function f, as if it was the x-value
• (g ∘ f)(x) = plug function f inside of function g, as if it was the x-value

Composite functions

• The domain of the inverse function of f(x) = the range of the function f(x)
• The range of the inverse function of f(x) = the domain of the function f(x)
• The inverse function of f(x) only exists if f is one-to-one on its given domains, otherwise it would be one-to-many, and that is not a function
• The graph of inverse f(x) is the reflection of f(x) in the line y=x

Transformation of graphs

• When one vertical and one horizontal transformation is applied, the order which it is applied does not matter
• When two vertical or two horizontal transformations are applied, the order which they are applied does matter: if dealing with horizontal transformations, you first translate, then stretch. If dealing with vertical transformations, you first stretch, then translate.
• Rules: