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:
  • 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 \(f(-x) \neq f(x)\) AND \(f(-x) \neq -f(x)\), the function is NEITHER ODD NOR EVEN