Relations, Functions and Inverses

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Relations, Functions, & Inverses:

Relations, Functions, & Inverses

Ordered Pair:

Ordered Pair An ordered pair is a set of 2 numbers— x and y— also known as coordinates of a point on a graph ( x , y )

Relation:

Relation A rule that gives an output (y) for each valid input (x) A set of ordered pairs For example: y = 3x {(- 2, -6), (1, 3), (0, 0),(4, 12)}

Function:

Function A relation where each input (x) only leads to one output (y) Any set of ordered pairs in which NO x value is repeated . * y values can be repeated For example: y = 5x 2 + 1 {(- 2, 21), (-1, 6), (0, 1),(4, 81)}

4 Different Graphical Representations:

4 Different Graphical Representations Of a function or relation…

1. Set Notation – Ordered Pairs:

1. Set Notation – Ordered Pairs {(-2,3),(0,-3),(2,-4),(7,0)} * ordered pairs encompassed by braces

2. Table of Values:

2. Table of Values x f(x) -2 3 -1 7 0 -1 7 6

3. Mapping Diagram:

3. Mapping Diagram

4. Graph:

4. Graph

The vertical line test:

The vertical line test In order to use the vertical line test, you must first plot the points on a coordinate plane. If you can draw a vertical line through each of the data points, and each line passes through EXACTLY one point, then the relation is a function.

Using the vertical line test.:

Using the vertical line test. This is a function. This is NOT a function. y x y x

Domain:

Domain The set of x values in a relation or function. In the relation: {(-2,3),(0,-3),(2,-4),(7,0)} t he domain is {-2, 0, 2, 7 } *If an x is repeated in the coordinates, you only write it once in the domain.

Range:

Range The set of y values in a relation or function. In the relation: {(-2,3),(0,-3),(2,-4),(7,0)} t he range is {-4, -3, 0, 3 } *If a y is repeated in the coordinates, you only write it once in the range.

Function Notation:

Function Notation f(x ) Read “f of x” Represents the function evaluated at x When you substitute a value in for x, it equals the value of y

Function Rule :

Function Rule An equation that represents a function f(x) = 6 x – 2 g(x) = -2x + 4

Synonyms for “Domain”:

Synonyms for “Domain” Domain Input x Independent variable

Synonyms for “Range”:

Synonyms for “Range” Range Output y Dependent variable

Inverse of a Relation:

I nverse of a Relation To find the inverse, switch each x and y For example: If the original, is (- 2, -6), (1, 3), (0, 0),(4, 12), the inverse would be (-6, - 2 ), (3, 1), (0, 0 ),(12, 4).

Inverse of a function:

Inverse of a function To find the inverse of a function, you need to switch x & y, and solve for y. Example: f(x) = 6 x – 2 Remember f(x) = y So, y = 6x – 2 Switch x & y x = 6y – 2 Now solve for y x + 2 = 6y Addition Property Division Property The inverse is which is also a function.  

Inverse Function Notation:

Inverse Function Notation If the inverse of a function is also a function, we use special notation So for the last example since y = is a function, we can rewrite it as .