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2.2 – Graphing Functions

Objectives

  • Graph and interpret functions.
  • Apply the vertical line test to identify functions.
  • Apply the horizontal line test to identify functions that are many-to-one.
  • Express and evaluate functions in piecewise notation, including greatest integer and absolute value functions.

 

Key Terms & Notes

Absolute Value Function
  • f(x) = |x|
  • The magnitude of x
  • This is a piecewise function
  • This is a Many-to-One function (crosses the graph more than once horizontally)
  • Written in piecewise form:

f(x) = \left\{\begin{array}{ll}-x, x\le0\\x, x\textgreater0\\\end{array}\right\}

  • Example of an Absolute Value Function graph with the horizontal line test

Alg2A-AbsoluteValueFx2

  • Another example of an Absolute Value function

Alg2A-AbsoluteValueFx

Equation: y=|x+4|-2

Dependent Variable
  • The y-value, output.
  • This is a variable in a function whose value is determined by the value of the independent variable.
    • y depends on x
  • The value of the dependent variable depends on the value of the Independent variable.
  • Graphed on the vertical axis, or y-axis, of the xy-plane.
Function
  • A special kind of relation in which each value of the input variable is paired with exactly one value of the output variable.
  • A special type of relationship in algebra where two quantities are related to each other so that one quantity depends on (or is a function of) the other.
  • A rule that tells how one number depends on another.
  • For each input number, there is exactly one output number.
  • Once you determine if a relation is a function, THEN you can decide if it is a one-to-one or a many-to-one function.
  • Passes the vertical line test.

Alg2A-VertLineFx

Alg2A-VertLineNotFx

Greatest Integer Function
  • The greatest integer less than or equal to x.
    • Look at the value of x, and look at the greatest integer that is less than or equal to the x value.
    • The y-value of the greatest integer function is always an integer (not a decimal).
    • This IS a piecewise function.
    • Written in the form: f(x) = [x]
      • Example: f(x) = [2.3].  The greatest integer less than or equal to 2.3 is 2.
      • Example: f(x) = [-6.7].  The greatest integer less than or equal to -6.7 is -7.
      • Example: f(x) = [3].  The greatest integer less than or equal to 3 is 3.
      • Example: f(3.6) = [[ 2x + 1 ]].  Substitute 3.6 for x, solve (answer = 8.2), then determine the greatest integer function (answer = 8).
      • Graph example of a Greatest Integer Function

Alg2A-Piecewise

 Equation: y=[x]+3

  • If any horizontal line can be drawn that intersects the graph at two or more points, then the inverse of the graph is not a function.
    • If no such line can be drawn, then the inverse of the graph is a function.
    • A graph’s inverse is when (x, y) coordinates are flipped / switched, then graphed.
  • This is a method for testing whether the inverse of a given graph is a function.
    • If no horizontal line intersects a graph at more than one point, the function is one-to-one.
Horizontal Line Test
  • A test to see if a function is one-to-one or many-to-one.
    • If it crosses a graph once, it is one-to-one.
    • If it crosses a graph more than once, it is many-to-one.
Independent Variable
  • The x-value, input.
  • This is a variable in a function that determines the value of the dependent variable.
    • Remember, y (the dependent) depends on x (the independent)
    • y is a function of (depends on) x.
  • Graphed on the horizontal axis, or x-axis, of the xy-plane.
  • Ex. Tree growth (dependent) is a function of sunlight (independent).
  • Ex. Babies (dependent) are a function of their mothers (independent).
Many-to-One Function
  • A function in which two or more x-values map to a given y-value.
    • This IS a function (because it passes vertical line test)
    • The horizontal line test shows it touches the function in MORE than one place, so it is a Many-to-One function.
      • Example: f(x) = 2x^3+2x^2-x
      • Example graph of a Many-to-One function: Alg2A-NotOnetoOne
Mapping Diagram
  • A diagram that shows how a specific relation maps input values to output values.

Alg2A 02.02 Mapping

One-to-One Function
  • A function in which each y-value has a single x-value mapped to it.
    • This IS a function (passes the vertical line test)
    • The horizontal line test shows it touches the function in ONLY one place, so it is a One-to-One function.
  • Example graph of a One-toOne function:

Alg2A-OnetoOne

Alg2A 02.02 1-to-1 Fx

Piecewise Functions
  • A functions whose domain is made up of two or more pieces (functions)
    • Each piece (or function) is defined over a portion of the original function’s domain.
    • Two common piecewise functions are:
      • Greatest Integer Functions
      • Absolute Value Functions
  • Example of a Piecewise function
    • f(x) = \left\{\begin{array}{ll}-x + 1, x\textless0\\-2, x=0\\x^2-1, x\textgreater0\end{array}\right\}
    • Answer: f(1)=0 and f(-1)=2 are both solutions
  • Example of a graph of a Piecewise function

Alg2A-GraphedFx

y = \left\{\begin{matrix}x^2+6, x\textless3\\-x+6, x\ge 3\end{matrix}\right.

Relation
  • A pairing of one set of information with another set of information.
    • A set of ordered pairs.
      • Domain: the first entry in each ordered pair
      • Range: the second entry in each ordered pair
    • If each element in the domain is paired with just one element in the range, the relation is called a function.
    • A relation can also be called a mapping.
Vertical Line Test
  • A method for testing whether a given graph is a function.
    • If a vertical line can be drawn that intersects the graph in two or more points, then the graph is not a function.
    • If no such line can be drawn, the graph is a function.

Alg2A 02.02 Graph Fx

 

Examples

Greatest Integer Function Example
  • Remember
    • The greatest integer function, shown below, is defined to be the greatest integer less than or equal to x
    • Written: f(x) = [[x]]
    • In a one-to-one function, each x-value corresponds to a different y-value.
    • In a many-to-one function, some x-values correspond to the same y-value.

 

  • Example
    • Andrew has a cell phone plan that provides 300 free minutes each month for a flat rate of $19. For any minutes over 300, Andrew is charged $0.39 per minute.
    • Which of the following piecewise functions represents charges based on Andrew’s cell phone plan?

f(x) = \left\{\begin{array}{ll}19, x\le300\\19+0.39(x-300), x\textgreater300\\\end{array}\right\}

    • The reason the 2nd expression has (x-300) is because the first 300 minutes are “free” or “included,” so you don’t want to pay $0.39 cents for any of those first 300 minutes.
      • So, you subtract them from that rate.
      • Then, all the minutes greater than 300 are charged at that rate.

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