6 Functions PDF

15.2 Relations, Functions and Inverses A relation is any: set of ordered pairs (co-ordinates); eg. graph; or rule, eg. FOUR TYPES of RELATIONS exist as shown below. ONLY TWO of these can also be classified as FUNCTIONS: One-to-One and Many-to-One. How do we tell which type? Th...

15.2 Relations, Functions and Inverses A relation is any: set of ordered pairs (co-ordinates); eg. graph; or rule, eg. FOUR TYPES of RELATIONS exist as shown below. ONLY TWO of these can also be classified as FUNCTIONS: One-to-One and Many-to-One. How do we tell which type? The H1V test can be used on any graph to quickly determine its relation classification. H1V stands for: Horizontal line test 1st , and then Vertical line test Visualise a horizontal line (drag it up and down over the graph) If it crosses the graph in any region more than once Many-to ……. relation. If horizontal line crosses the graph only once, One-to- ……. relation. Now visualise a vertical line (drag it left and right across the graph) If it crosses the graph in any region more than once, ……….-to-Many relation. if a vertical line crosses the graph only once, ………. -to-One relation. Combination of the horizontal and vertical line test results will enable the relation type to be found. Can then conclude if relation is also a function. Function Notation Consider the relation , which is a function. The -values are determined from the -values, so we say ‘ is a function of ’, which is abbreviated to and said ' equals f of '. So, a rule can also be written in function notation as Similarly, a rule can also be written in function notation as. This is said " of equals pi squared". Further to this, for a given function = , the value of when =1 is written as…. (1), the value of when =5 is written as…. (5), the value of when = as ( ), etc. So from the above example, " refers to the area of a circle of radius 2 and is the singular dependent 'y' value,. NOTE: Context when reading maths is clearly very important as typically, would mean A multiplied by 2; which it most definitely does NOT when A represents a function. Similarly, does NOT mean multiplied by but to repeat, means is a function of or more simply, represents the "y variable". Worked Example: Let 6 Functions Page 1 when A represents a function. Similarly, does NOT mean multiplied by but to repeat, means is a function of or more simply, represents the "y variable". Worked Example: Let Evaluate: Solve: Simplify: Domain and Range of a relation The domain of a relation, is the set of 'allowable' inputs ('x' values) such that output(s) ('y' values) can be evaluated] If the domain is not explicitly given, it is implied to be the largest subset of the real number set, , for which the rule will have defined “y-coordinate” values. For example: a. If maximal (implied) domain is b. If maximal (implied) domain is c. If maximal (implied) domain is or The range of a relation, is the set of permissible output(s) ('y' values) from the rule/graph/set of ordered pairs. (Unless this is linear, do not assume substituting domain endpoints will find the maximum and minimum y values and thus your range). Inverse Relation An inverse is found by interchanging x and y variables of any given relation. It acts to 'undo' the series of operations on a variable. Note: Not to be confused with an inverse relationship, which simply refers to one in which as one variable increases, the other decreases (or vice versa). Inverse FUNCTIONS We use the notation, to represent the inverse function, or just BEWARE: For a function to have an inverse that is also a function, the original function must be one -to-one. See example below. A many-to-one function such as a parabola will have an inverse relation (one-to-many), unless a restricted domain is placed on the original function to force the original function to become a one-to-one function. 6 Functions Page 2  the original function to force the original function to become a one-to-one function. Only one-to-one functions, f, will have inverse functions, f -1 Finding the rule for an inverse function: 1. Interchange x & y variables from original rule 2. Rearrange to make y the subject 3. Take extra care with algebra to select the rule with the appropriate range 4. Ensure the domain for the inverse is given Worked Examples: Determine the equation for the inverse functions for the following rules. If necessary, apply a restriction to the function s o it has an inverse function. 1. 2. 3. 1. 6 Functions Page 3 DrT Transformations Summary 6 Functions Page 4 Do Desmos Card Sort Activity - Transformations follow link Activity link: https://student.desmos.com/activitybuilder/student- greeting/6695c5ac1026c1f0d5a3d3c4 If you do not have account/link to 10F class yet: https://student.desmos.com/? prepopulateCode=d9pych 6 Functions Page 5 9.8 Rectangular Hyperbolas: Key Features Revised z TRANSLATIONS (in x axis and y axis) REFLECTION in x-axis REFLECTION in y-axis DILATIONS (from x axis by factor a) CAN COMBINE ALL ABOVE Worked Examples Sketch the graph of 6 Functions Page 6  9.6 Exponential Graphs BASIC equation: where is a constant (parameter) and more formally, is a positive real number excluding zero (or ) Or and more common variables, as = number of bacteria/people etc. = time (seconds/minutes/years etc.) More complex/transformed equation: where A, a, b and c are all constants (parameters) , stretches/compresses graph from the x-axis by factor and can reflect graph in x-axis if