Section 3.4 Transformations of Functions Objectives 1-6: Using Transformations to Graph Functions

Vertical Shifts of Functions If c is a positive real number: The graph of ( )y f x c= + is obtained by shifting the graph of ( )y f x= vertically upward c units. The graph of ( )y f x c= − is obtained by shifting the graph of ( )y f x= vertically downward c units.

Horizontal Shifts of Functions If c is a positive real number: The graph of ( )y f x c= + is obtained by shifting the graph of ( )y f x= horizontally to the left c units. The graph of ( )y f x c= − is obtained by shifting the graph of ( )y f x= horizontally to the right c units. Note: For 0c > , the graph of ( )y f x c= − is the graph of f shifted to the right c units. At first glance, it appears that the rule for horizontal shifts is the opposite of what seems natural. Substituting x c+ for x causes the graph of ( )y f x= to be shifted to the left while substituting x c− for x causes the graph to shift to the right c units. 3.4.2, 3.4.13 Use the graph of a basic function and a combination of transformations to sketch the functions. 2. y = x + 2 13. k(x) = x −1

Stretches and Compressions of Functions Supposea is a positive real number: The graph of ( )y af x= is obtained by the multiplying each y-coordinate of ( )y f x= by a . If 1a > , the graph of ( )y af x= is a vertical stretch of the graph of ( )y f x= . If 0 1a< < , the graph of ( )y af x= is a vertical compression of the graph of ( )y f x= . 3.4.43 Use the graph of a basic function and a combination of transformations to sketch the function. g(x) = 3x2 Reflection of Functions about the x-Axis The graph of ( )y f x= − is obtained by reflecting the graph of ( )y f x= about the x-axis. Reflections of Functions about the y-Axis The graph of ( )y f x= − is obtained by reflecting the graph of ( )y f x= about the y-axis.

When sketching a function that involves multiple transformations it is important to follow a certain “order of operations”. Below is the order in which each transformation will be performed in this text (Remember: HSRV)

1) Horizontal Shifts 2) Stretches/Compressions 3) Reflection about y-axis 4) Reflection about x-axis 5) Vertical Shifts

Different ordering is possible for transformations 2) through 5), but you should always perform the horizontal shift first and the vertical shift last. 3.4.25, 3.4.32, 3.4.54 Use the graph of a basic function and a combination of transformations to sketch the functions. 25. g(x) = −x2 − 2 32. g(x) = (3− x)2

54. y = 1x −3

+ 2

Section 3.5 The Algebra of Functions; Composite Functions In this section we will learn how to create a new function by combining two or more existing functions. First we will combine functions by adding, subtracting, multiplying or dividing two existing functions. The addition, subtraction, multiplication and division of functions is also known as the algebra of functions. The Algebra of Functions Let f and g be functions. For all x such that both ( )f x and ( )g x are defined, the sum, difference, product, and quotient of f and g exist and are defined as follows:

1. The sum of f and g : ( )( ) ( ) ( )f g x f x g x+ = +

2. The difference of f and g : ( )( ) ( ) ( )f g x f x g x− = −

3. The product of f and g : ( )( ) ( ) ( )fg x f x g x=

4. The quotient of f and g : ( ) ( )for all ( ) 0

( )f f x

x g xg g x

! "= ≠$ %

& '

Objective 1: Evaluating a Combined Function 3.5.2. If f (x) = x +1 and g(x) = x2 − 2 , find ( f − g)(8) . Objective 3: Finding Combined Functions and their Domains For a number a to be in the domain of f g+ , a must be in the domain of f and a must be in the domain of g. In other words, the domain of f g+ must be the intersection of the two domains. (Recall that the intersection A BI of two sets of real numbers A and B is the set of all values that A and B have in common)

The Domain of , , and ff g f g fgg

+ −

Suppose f is a function with domain A and g is a function with domain B then:

1. The domain of the sum, f g+ , is the set of all x in A BI . 2. The domain of the difference, f g− , is the set of all x in A BI . 3. The domain of the product, fg , is the set of all x in A BI .

4. The domain of the quotient, fg

, is the set of all x in A BI such that ( ) 0.g x ≠

3.5.21 Find f+g, f – g, fg, f/g. Determine the domain for each function. f (x) = x2 + 2 , g(x) = x −1

Objective 4: Forming and Evaluating Composite Functions Definition The Composite Function Given functions f and ,g the composite function, f go (also called the composition of f and g ) is defined by

( ) ( )( ) ( )f g x f g x=o provided ( )g x is in the domain of f. The composition of f and g does not equal the product of f and g : ( )o ( ) ( )f g x fg x≠ .

Also , the composition of f and g does not necessarily equal the composition of g and f though this equality does exist for certain pairs of functions.

Let f (x) = 3x +1 , g(x) = 2x +1

, h(x) = x +3

3.5.33. Find the function f ! g. 3.5.48. Find (g !h)(−2)

Objective 5: Determining the Domain of Composite Functions Suppose and f g are functions. For a number x to be in the domain of f go , x must be in the domain of g and ( )g x must be in domain of f.

To find the domain of f go :

1. Find the domain of g . 2. Exclude from the domain of g all values of x for which ( )g x is not in the domain of f.

3.5.61 For the given functions find the domain of f ◦g and g ◦f. f (x) = x2 , g(x) = x

3.5.66 For the given functions find the domain of f ◦g and g ◦f.

f (x) = 1x − 2

, g(x) = 4− x