Domain - the set of all possible ( ) values of a relation. Range - the set of all possible ( ) values of a relation.

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1 Definitions: Domain - the set of all possible ( ) values of a relation. Range - the set of all possible ( ) values of a relation. Relation - a set of ordered pair(s) Pre- Calculus Mathematics Functions and Relations Goals: 1. Identify relations, functions, one- to- one functions, domains, ranges, vertical and horizontal line tests, restrictions. Recognize function types Function - a relation in which each domain ( ) value is paired with only one unique range ( ) value. Vertical line test - an equation defines y as a function of if and only if every vertical line in the coordinate plane intersects the graph of the equation only once. Eample 1: Determine the domain/range of the following graphs and whether they are a function/relation Restriction on the domain of a function: 1. Cannot have a negative number inside an even root. f( ) = 3. Cannot have zero in a denominator 8 f( ) = 6

2 Pre- Calculus Mathematics Functions and Relations One to One Function A one- to- one function is a function in which every single value of the domain is associated with only one value in the range, and vice- versa. Horizontal line test - for a one- to- one function: A function, f(), is a one- to- one function of if and only if every horizontal line in the coordinate plane intersects the function only once at most. Eample : Determine whether the following relations are functions, one- to- one functions or neither A 1 A 1 A A 4 B B B B 6 3 C 3 C 1 C 8 C 4 D 4 D D 10 D 5 E 5 E E Practice: Pg 9 & 10 # 1, and 3

3 Pre- Calculus Mathematics Arithmetic Combinations of Functions Goal: Perform operations with functions Functions are number generators. When you put a value for the domain in the function, you will get the resulting value in the range. f = + 5 ( ) And just like numbers, functions can be added, subtracted, multiplied and divided. f = 1 & g = 4 to determine: Use the following functions, ( ) ( ) a) Sum: ( f + g)( ) = f() + g() b) Difference: ( f g)( ) = f() g() f g Product: ( fg)( ) = f ( ) g( ) Quotient: ( ) = f( ) g( ) Note: The domain of the new function must include the restrictions of the new functions as well as the restriction(s) of the original function(s) Eample 1: Given functions below, determine each new combined function and its domain. f 1 g + h ( ) =, ( ) =, ( ) = 5 3 i ( ) = 9, j ( ) =, k ( ) = f g a) ( gj)( 3) b) ( 4)

4 Pre- Calculus Mathematics Arithmetic Combinations of Functions Eample 1 continued Given functions below, determine each new combined function and its domain. f 1 g + h ( ) =, ( ) =, ( ) = 5 3 i ( ) = 9, j ( ) =, k ( ) = gk i c) ( g f )( ) d) ( ) 1 1 k k e) ( ) f f) ( )

5 Pre- Calculus Mathematics Arithmetic Combinations of Functions Eample 1 continued Given functions below, determine each new combined function and its domain. f 1 g + h ( ) =, ( ) =, ( ) = 5 3 i ( ) = 9, j ( ) =, k ( ) = g h f i g) ( ) ( ) h) g( h i) ( ) Practice: Pg # 1,, 3 (a, b, c, d, i, j), 4 (a, d)

6 Goals: 1. Perform the composite of two or more functions. Domain of a composite function Pre- Calculus Mathematics Composite Functions A function can be considered like a machine. It has an input value () and generates an output, y, or f( ). ( ) = 3 4 f A function can also be input into another function. This would then generate a composite function. f = 3 4 & g = + Using the functions ( ) ( ) Determine: ( ( )) ( ) f g g f ( ) Notation: ( ) ( ) ( )( ) f g = f og = f o g With composite functions, the output of one function becomes the input of another function(s). When determining ( f og)( ) the output of function gbecomes ( ) the input of the function f( ). Eample 1: Given functions below, determine each composite function. f 1 g + h ( ) =, ( ) =, ( ) = 5 3, i ( ) = 9, k ( ) = a) ( ho g)( 3) b) ( kogo f)( )

7 Pre- Calculus Mathematics Composite Functions Eample 1 continued. f 1 g + h ( ) =, ( ) =, ( ) = 5 3, i ( ) = 9, k ( ) = c) ( h i)( ) o d) i( h( )) Domain of a Composite Function The domain of ( f og)( ) has both the original restriction(s) of gas ( ) well as the domain restrictions of the final composite function ( f o g)( ). Eample : Given the functions and its domain. 4 f( ) =, g( ) = 3 determine each composite function a) ( f o g)( ) b) ( go f)( ) c) ( go g)( )

8 Pre- Calculus Mathematics Composite Functions Eample 3: Calculus Application: Compute f h f h ( + ) ( ), for ( ) 3 = f Practice: Pg 6 #1 (a, b, c, d), (a, b, g), 3(a, f), 4(a, b, c, d, e)

9 Pre- Calculus Mathematics Transformations of Graphs Part 1 Goals: 1. Identify and apply a vertical translation to a function with or without a graph. Identify and apply a horizontal translation to a function with or without a graph Vertical translations (shifting the graph up or down) form: y = f h + k Square Root Absolute Value y = y = + 3 So for y = f h + k If k is positive à If k is negative à Eample 1: Given the graph of y = f below, describe the transformation applied graph y = f, and map the coordinates of the image points.

10 Pre- Calculus Mathematics Transformations of Graphs Part 1 Horizontal Translations (shifting the graph left or right) form: y = f h Graph: y =! ; y = 3! ; y = +! For y = f h If h is positive à If h is negative ß Eample : Given the graph of y = f below, describe the transformation applied graph y = f, and map the coordinates of the image points.

11 Pre- Calculus Mathematics Transformations of Graphs Part 1 Eample 3: Given the graph of y = f below, describe the transformation applied graph y = f 3 4, and map the coordinates of the image points. Eample 5: Given the function y = f below, describe the transformation applied to each of the functions below. a) y = f + 3 b) y = f c) y + 1 = f + 5 d) y 4 = f 7 7 Practice: Complete the questions on the net page, the solutions are provided

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13 Pre- Calculus Mathematics Transformations of Graphs Part Goals: 1. Identify and apply a reflection to a function with or without a graph. Identify and apply a vertical compression/epansion to a function 3. Identify and apply a horizontal compression/epansion to a function Reflections in the Coordinate Ais (flipping the graph over the or y ais) y = y = So for y = f( ) If y = f( ) à If y = f( ) à Eample 1: Given the graph of y f( ) = below, describe the transformation applied to the graph y = f( ), and map the general coordinates of the image points.

14 Pre- Calculus Mathematics Transformations of Graphs Part Vertical Epansions/Compressions (stretching the graph in y direction) Graph: y = y = y = 1 For y = af( ) If a > 1 à If a < 1 à Eample : Given the graph of y = f( ) below, describe the transformation applied to the graph 1 y = f ( ), and map the general coordinates of the image points. 3

15 Pre- Calculus Mathematics Transformations of Graphs Part Horizontal Compressions/Epansions (stretching the graph in direction) Graph: y = ( ) y= 1 y = For y = f( b) If b > 1 à If b < 1 à Eample 3: Given the graph of y = f( ) below, describe the transformation applied to the graph y= f( ), and map the general coordinates of the image points.

16 Pre- Calculus Mathematics Transformations of Graphs Part Eample 4: Given the function y = f( ) below, describe the transformation applied to each of the functions below. a) y= f( 4) + 6 b) y= f( 3 1) 4 c) y+ 10 = f(5 ) Eample 5: Given the graph of y = f( ), sketch the graph of 1 1 y = f 3 Eample 6: If (- 3, 4) is a point on the graph y = f( ) what must be the point on the graph y+ 7= 5 f( 4) + 3. Practice: Pg 34 # 1 (a, c, e), (a, c, e, f), 3, 4, 6, 9c, 10 (b, d, f), 11d, 1(c, g)

17 Goals: 1. Determine the inverse of a given function. Determine whether two functions are inverses 3. Graph the inverse of a function Pre-Calculus Mathematics Inverse Functions Two functions are inverse functions if one function undoes what the other function does. Inverse functions are just a reflection across the line y =. In order for both a function f( ) and its inverse f 1 ( ) to qualify as functions, f( ) must be a one-to-one function. Determining the inverse of a function: 1. Verify that f () is a one-to-one function. (If not, its inverse is not a function.). Replace f () with y and echange all s and y s. 3. Solve for y. 4. Replace the new y with f 1 ( ). (Only if f 1 ( ) is actually a function.) Eample 1: If f ( ) 3, determine the inverse of the function. Eample : If g ( ), determine the inverse of the function. 3 1

18 Eample 3: If h ( ) 3 Pre-Calculus Mathematics Inverse Functions, determine the inverse of the function. Determining if two functions are inverses: ( undo each other) Two functions f ( ) and g( ) are inverses of each other if and only if f g ( ) for every value of in the domain of g and g f ( ) for every value of in the domain of f. Eample 4: Determine whether f( ) 3 and 3 g ( ) are inverses of each other. 1

19 Pre-Calculus Mathematics Inverse Functions Graphing the inverse of a function The graph of a function and its inverse are symmetric about the line y. f( ) is the reflection of f 1 ( ) on the line y and vice versa. f Eample 5: Given the function whether the inverse of the function is a function. ( ) 4, graph the inverse of the function. Determine Eample 6: If (-3, 4) is a point on the graph y f ( ) what must be the point on the graph y f 1 1 ( ). Practice: Page 44 # (a,b,c), 4, 5(a,b,f), 6 (a, c, f), 7(a, b, c)

20 Pre- Calculus Mathematics Combined Transformations Goal: Perform reflections, epansions, compressions and translations of a function both with and without a graph. All of the transformations we have performed can be summarized as follows: y = f( ) transforms to y= af [ b( h) ] + k Reflections: Epansion: a < 0, reflection in the - ais b < 0, reflection in the y- ais 1 f reflection across the line y = a > 1, vertical epansion by a factor of a b < 1, horizontal epansion by a factor of 1 b Compression: a < 1, vertical compression by a factor of a b > 1, horizontal compression by a factor of 1 b Translation: k > 0, vertical translation k units up k < 0, vertical translation k units down h > 0, horizontal translation h units right h < 0, horizontal translation h units left When combining transformations, the reflections/epansions/compressions must occur before the translations. Eample 1: Given point P ( 4, ) on y = f( ) find the new location for P on: y = f + 3 Eample : Given point P (a, b) on y = f( ) find the new location for P on: ( ) y= 3f 6 5

21 Eample 3: Graph the following functions: Pre- Calculus Mathematics Combined Transformations a) 1 y= b) h ( ) ( ) = c) If f ( ) 1 = = +, graph y f ( )

22 Pre- Calculus Mathematics Combined Transformations Eample 4: Given the graph of y = f( ), sketch the graph y = f( + 3) + 1 Practice: Page 51 # (a, b, c, e, g, j), 3(a, b, c, d), 6 (a, b, c), 7(a, c, e)