Mathematics: Functions
Definition
- A function is a relation between two sets of data where each input has 1 or less potential outputs
- Horizontal Lines, Parabolas, Linear Equations, Hyperbolas, Exponentials, Polynomials and Cubic Graphs are all examples of functions
- Circles and Vertical Lines are NOT functions
- In other words, functions can be one-to-one or many-to-one relationships, but not one-to-many relationships (In reference to input and output values)
Notation
- There are 3 methods of expressing functions:
- All of the above methods say the same thing:
- When
is the input, is the output
- When
- For example:
- All state that when
is the input, is the output
Vertical Line Test
- The vertical line test is a quick way to test if a graph is a function
- If a vertical line can cut the function TWICE OR MORE, the graph is not a function
- In the graph below, the red graph is a function, but the blue line is not, because the green vertical line cuts the blue line at 2 points
Set Notation
- In set notation, different types of brackets have different meanings:
- “(” and “)” are used to write a set where the boundaries are EXCLUDED
- “[” and “]” are used to write a set where the boundaries are INCLUDED
means Infinity while means Negative Infinity means that “ is in the set of all numbers between 1 and infinity”
Domain And Range
- All functions have a Domain and Range
- The domain of a function is all the valid input values
- The range of a function is all the valid output values
- Some input values are INVALID and therefore not part of the Domain
- For Example:
- In
, only positive values of are possible (because negative numbers have no graphable roots) - Therefore,
must be greater than or equal to zero (0) - This can be expressed as
OR
- In
- For Example:
- Some output values are INVALID and therefore not part of the Range
- y-asymptotes are not part of the range
- All y values above/below the minimum/maximum y of a graph are not part of the range
Transformations of a Function (from )
- Vertical Translation Up
units: - Vertical Translation Down
units: - Horizontal Translation Left
units: - Horizontal Translation Right
units:
Odd and Even functions
- Even Functions:
- Symmetrical about the y-axis
- Rules:
- If
is a valid solution to , is in the same function
- Odd Functions:
- Symmetrical about the origin
- Rules:
- If
is a valid solution to , then is also a valid solution
- Symmetrical about the origin
- Proving/Solving Odd and Even Functions:
- Find
- Simplify
- If
, the function is ODD - If
, the function is EVEN - If
AND , the function is NEITHER ODD NOR EVEN
- Find