Absolute Value Function 
 f(x) = x
 The magnitude of x
 This is a piecewise function
 This is a ManytoOne function (crosses the graph more than once horizontally)
 Written in piecewise form:
 Example of an Absolute Value Function graph with the horizontal line test
 Another example of an Absolute Value function
Equation:

Dependent Variable 
 The yvalue, output.
 This is a variable in a function whose value is determined by the value of the independent variable.
 The value of the dependent variable depends on the value of the Independent variable.
 Graphed on the vertical axis, or yaxis, of the xyplane.

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 onetoone or a manytoone 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 yvalue 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:
 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 onetoone.

Horizontal Line Test 
 A test to see if a function is onetoone or manytoone.
 If it crosses a graph once, it is onetoone.
 If it crosses a graph more than once, it is manytoone.

Independent Variable 
 The xvalue, 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 xaxis, of the xyplane.
 Ex. Tree growth (dependent) is a function of sunlight (independent).
 Ex. Babies (dependent) are a function of their mothers (independent).

ManytoOne Function 
 A function in which two or more xvalues map to a given yvalue.
 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 ManytoOne function.
 Example:
 Example graph of a ManytoOne function:

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

OnetoOne Function 
 A function in which each yvalue has a single xvalue 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 OnetoOne function.
 Example graph of a OnetoOne 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
 Answer: and are both solutions
 Example of a graph of a Piecewise function

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.
