# 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:
- \(y=123\)
- \(f(x)=123\)
- \(f:x→123\)

- All of the above methods say the same thing:
- When \(x\) is the input, \(123\) is the output

- For example:
- \(y=2x\)
- \(f(x)=2x\)
- \(f:x→2x\)

- All state that when \(x\) is the input, \(2x\) 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**

- “(” and “)” are used to write a set where the boundaries are
- \(\infty\) means Infinity while \(- \infty\) means Negative Infinity
- \(x\in[1,\infty)\) means that “\(x\) 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 \(g(x)=\sqrt{x}\), only positive values of \(x\) are possible (because negative numbers have no graphable roots)
- Therefore, \(x\) must be greater than or equal to zero (0)
- This can be expressed as \(x \geq 0\) OR \(x\in(0,\infty)\)

- 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 \(f(x)\))

- Vertical Translation Up \(c\) units: \(f(x)+c\)
- Vertical Translation Down \(c\) units: \(f(x)-c\)
- Horizontal Translation Left \(c\) units: \(f(x+c)\)
- Horizontal Translation Right \(c\) units: \(f(x-c)\)

## Odd and Even functions

- Even Functions:
- Symmetrical about the y-axis
- Rules:
- \(f(-x)=f(x)\)
- If \((x,y)\) is a valid solution to \(f(x)\), \((x,-y)\) is in the same function

- Odd Functions:
- Symmetrical about the origin \((0,0)\)
- Rules:
- \(f(-x)=-f(x)\)
- If \((x,y)\) is a valid solution to \(f(x)\), then \((-x,-y)\) is also a valid solution

- Proving/Solving Odd and Even Functions:
- Find \(f(-x)\)
- Simplify \(f(-x)\)
- If \(f(-x) = -f(x)\), the function is ODD
- If \(f(-x) = f(x)\), the function is EVEN
- If \(f(-x) \neq f(x)\) AND \(f(-x) \neq -f(x)\), the function is NEITHER ODD NOR EVEN