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:
  • $y=123$
  • $f(x)=123$
  • $f:x\to 123$
• All of the above methods say the same thing:
  • When $x$ is the input, $123$ is the output
• For example:
  • $y=2x$
  • $f(x)=2x$
  • $f:x\to 2x$
• 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
• $\mathrm{\infty }$ means Infinity while $-\mathrm{\infty }$ means Negative Infinity
• $x\in [1,\mathrm{\infty })$ 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 $x\ge 0$ OR $x\in (0,\mathrm{\infty })$
• 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:
  • 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:
  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 $f(-x)\ne f(x)$ AND $f(-x)\ne -f(x)$, the function is NEITHER ODD NOR EVEN