Lessons 1 and 2: Functions and Evaluating Functions

• What is a relation in mathematics?

  A relation is any set of ordered pairs.
• What is the domain of a relation?

  The domain is the set of all first elements of the ordered pairs.
• What is the range of a relation?

  The range is the set of all second elements of the ordered pairs.
• How does a function differ from a general relation?

  A function corresponds each element in the domain to exactly one element in the range.
• What are the four ways to determine if a relation is a function?

  1. Ordered Pairs
  2. Table of Values
  3. Mapping Diagram
  4. Vertical Line Test
• Correspondences of Functions

  FUNCTION:

  1. One-to-one - no domain and range are repeated.
  2. Many-to-one - only the range is repeated.

  NOT FUNCTION:

  1. One-to-many - only the domain is repeated.
  2. Many-to-many - both the domain and range are repeated.
• Function Machine


  A function can be illustrated as a machine where there is the input and the output.
• Piecewise Function


  A function in which more than one formula is used to define the output.
• Types of Functions

  1. Constant Function
  2. Identity Function
  3. Polynomial Function
  4. Linear Function
  5. Quadratic Function
  6. Cubic Function
  7. Power Function
  8. Rational Function
  9. Exponential Function
  10. Logarithmic Function
  11. Absolute Value Function
  12. Greatest Integer Function
• Constant Function

  • A constant function has the same output value no matter what your input value is.
  • $f(x)=$$b$
• Identity Function

  • The identity function returns the same value and uses as its argument.
  • $f(x)=$$x$
• Polynomial Function

  • Defined by $y =$$a^0 +$$a^1x +$$a^2x^2 +$$... +$$a^nx^n$
• Linear Function

  • The polynomial function with degree one.
  • $y =$$mx +$$b$
• Quadratic Function

  • If the degree of the polynomial function is two, then it is a quadratic function.
  • $y =$$ax^2 +$$bx +$$c$
• Cubic Function

  • A cubic polynomial function is a polynomial of degree three.
  • $y =$$ax^3 +$$bx^2 +$$cx +$$d$
• Power Function

  • Many of our parent functions such as linear and quadratic functions are functions.
  • $y =$$ax^b$
• Rational Function

  • A rational function can be represented by a rational fraction.
  • $p(x)/q(x)$
• Exponential Function

  • This function is in the form $y =$$ab^x$, where x is an exponent and a and b are constants. (Note: only b is raised to the power x; not a.) If the base b is greater than 1, then the result is exponential growth.
• Logarithmic Function

  • Inverses of exponential functions and vice versa.
  • Very useful in permitting us to work with very large numbers while manipulating numbers of a much more manageable size.
  • $y =$$log_{b} x$
• Absolute Value Function

  • Represented in the form: $f(x) =$$|x|$
  • "Always positive"
• Greatest Integer Function

  • Defined by the form: $f(x) =$$||x||$
  • Remember that no matter how large the decimal of the number inside the greatest integer sign, it will always round down.
• Evaluating Functions


  Is the process of determining the value of the function at the number assigned to a given variable.