# 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

• Another example of an Absolute Value function

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.

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

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:
Mapping Diagram
• A diagram that shows how a specific relation maps input values to output values.

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:

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

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

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.

## 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.