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:x123
  • 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:x2x
  • 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
  • means Infinity while means Negative Infinity
  • x[1,) 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)=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 x0 OR x(0,)
  • 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(xc)

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)f(x) AND f(x)f(x), the function is NEITHER ODD NOR EVEN


Pranav Sharma
Pranav Sharma
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