# 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
• $$\infty$$ 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 $$x\in(0,\infty)$$
• 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:
• 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:
1. Find $$f(-x)$$
2. Simplify $$f(-x)$$
3. If $$f(-x) = -f(x)$$, the function is ODD
4. If $$f(-x) = f(x)$$, the function is EVEN
5. If $$f(-x) \neq f(x)$$ AND $$f(-x) \neq -f(x)$$, the function is NEITHER ODD NOR EVEN

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