AuthorsW11: Keri Austhof Caleb Medacco Stephanie KladderF11: Greg O'Hearn, Jeremy Sheaffer, and Shannon Penix

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~Important Definitions to Know~
  • Function:

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-A relationship between two variables, typically x and y, is called a function if there is a rule that assigns to each value of x
one and only one value of y. We say then that y is a function of x.

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A function f takes an input, x, and returns an output f(x). One metaphor describes the function as a "machine" or "black box" that converts the input into the output. A function, in a mathematical sense, expresses the idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns exactly one value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from an given sets: the domain and the codomain of the function. An example of a function with the real numbers as both its domain and codomain is the function f(x) = 2x, which assigns to every real number the real number with twice its value. In this case, it is written that f(5) = 10.


  • Variable- A symbol for a number we don't know yet. It is usually a letter like x or y.- It is also a symbol that represents a quantity in an algebraic expression or it can also be a
value that may change within the scope of a given problem or set of operations.
- The variables of a function are typically the symbols ' x ' and ' y ' but are not limited to these two symbols.

  • Independent Variable

-A variable whose value determines the value of other variables
-Also a variable in a mathematical equation or statement whose value determines that of the dependent variable: in y = f(x), x is the independent variable

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  • Dependent Variable

-A mathematical variable whose value is determined by the value assumed by an independent variable
-Also a variable in a mathematical equation or statement whose value depends on that taken on by the independent variable
in "y=f(x)", "y" is the dependent variable

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  • Domain

-The set of all possible values of an independent variable of a function.
-For a broader definition and more information about domain visit domain at Wolfram|Alpha

  • Range

-The set of all values a given function may take on
-For a broader definition and more information about range visit range at Wolfram|Alpha


~History of Functions~
The idea of a function was developed in the seventeenth century. During this time, Rene Descartes (1596-1650), in his book Geometry (1637), used the concept to describe many mathematical relationships. The term "function" was introduced by Gottfried Wilhelm Leibniz (1646-1716) almost fifty years after the publication of Geometry. The idea of a function was further formalized by Leonhard Euler (1707-1783) who introduced the notation for a function, y = f(x).
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Left: Gottfried Wilhelm Leibniz

Middle: Rene Descartes

Right:Leonhard Euler

~Ways to Represent a Function~
  • Symbolic

    • This is also referred to the algebraic form of representing a function and is often the most common form to do so.
    • Often, in mathematics, you will work with more than one function and therefore names are given to functions. The basic name given to any function is either f or g.
    • f (x) = y
      • This is read: "f of x equals y"
      • This means: If we would input x into our function f what would come out is the value y.
  • Table

    • A table is a way to represent the pair of input values (x) with the output values (y).
    • For example: The table for f(x) = 2x + 1

  • Graph:

    • Asymptote
      • An asymptote is a line that the graph of a function approaches, but never intersects.
      • An asymptote can occur when a denominator in a function includes a variable that cannot be canceled out by something in the numerator.
      • Horizontal asymptotes are horizontal lines that the graph of a function approaches as x tends to approach positive or negative infinity.
      • Vertical asymptotes are vertical lines near which the function grows to infinity.
      • When a linear asymptote is not horizontal or vertical, it is called an oblique or slant asymptote. This type of asymptote occurs when the numerator of a rational function is exactly one degree greater than the denominator. Determining the asymptotes of a function is an important step in sketching its graph.
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        • Vertical-Line Test
          • Vertical Line Test is a test used to determine whether a relation is a function or not. A graph is said to be a function if the vertical line drawn does not intersect the graph more than one point.
          • Using vertical line test, the graph given represents a function as the vertical line drawn intersects the graph at only one point.
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                • Using vertical line test, the graph given does not represent a function as the Vertical line drawn intersects the graph at more than one point.

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          • It is important to not that these three ways to represent a function does not mean that there are three different types of functions. All of these representations can describe one function, they are just different ways to look at it.


          • ~Standards for Functions~
          • Interpreting Functions**
              • Understand the concept of a function and use function notation
              • Interpret functions that arise in applications in terms of the context
              • Analyze functions using different representations

            • Building Functions
              • Build a function that models a relationship between two quantities
              • Build new functions from existing functions

            • Linear, Quadratic, and Exponential Models
              • Construct and compare linear and exponential models and solve problems
              • Interpret expressions for functions in terms of the situation they model

            • Trigonometric Functions
              • Extend the domain of trigonometric functions using the unit circle
              • Model periodic phenomena with trigonometric functions
              • Prove and apply trigonometric identities

            • Mathematical Practices
              1. Make sense of problems and persevere in solving them.
              2. Reason abstractly and quantitatively.
              3. Construct viable arguments and critique the reasoning of others.
              4. Model with mathematics.
              5. Use appropriate tools strategically.
              6. Attend to precision.
              7. Look for and make use of structure.
              8. Look for and express regularity in repeated reasoning.

          • Common Core High School Function Standards for Linear, Quadratic, and Exponential Models:

          • Construct and compare linear, quadratic, and exponential models and solve problems.

          • Distinguish between situations that can be modeled with linear functions and with exponential functions.

          • Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

          • Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

          • Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

          • Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

          • Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

          • For exponential models, express as a logarithm the solution to abct = d where a, c, and dare numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
          • Interpret expressions for functions in terms of the situation they model.

          • Interpret the parameters in a linear or exponential function in terms of a context.

          • ~Applications~
          • A very important question that all of you should be asking yourself is why do we need functions? What makes them so great and important in the math world?
          • READ THIS!

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          • This link also has some great other articles concerning functions and the reasons to why we use them.

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