Chapter 1: Section 1.3

New Functions and Old Functions

Transformations:Vertical and Horizontal Shifts: Suppose c > 0. To obtain the graph of

y = f(x) + c, shift the graph of y = f(x) a distance of c units .

y = f(x)� c, shift the graph of y = f(x) a distance of c units .

y = f(x� c), shift the graph of y = f(x) a distance of c units .

y = f(x+ c), shift the graph of y = f(x) a distance of c units .

Vertical and Horizontal Stretching and Reflecting: Suppose c > 1. To obtain the graph of

y = cf(x), the graph of y = f(x) by a factor of c.

y = (1/c)f(x), the graph of y = f(x) by a factor of c.

y = f(cx), the graph of y = f(x) by a factor of c.

y = f(x/c), the graph of y = f(x) by a factor of c.

y = �f(x), the graph of y = f(x) .

y = f(�x), the graph of y = f(x) .

Chapter 1: Sec1.3, New Functions and Old Functions

Example 1: Starting with the function f(x), list the transformations for the following.

(a) y = f(2x) (b) y = f(1

2x)

(c) y = �f(x) (d) y = f(�(x� 1))

2 Spring 2017, Maya Johnson

( b ) Stretching horizontally by a factor of 2 .

qreee.EE?oMEtCYiIIiIoafaeto.

Feting about the y- axis .

Chapter 1: Sec1.3, New Functions and Old Functions

Example 2: List the transformations of y = x

2 that result in g(x) = �(x� 2)2 + 9.

Example 3: Sketch the graph of the function g(x) = �(x� 2)2 + 9. (Note; When graphing parabolas,

clearly indicate where the vertex, x-intercept and y-intercept are on the graph)

3 Spring 2017, Maya Johnson

Chapter 1: Sec1.3, New Functions and Old Functions

Combination of Functions

Let f and g be two functions with domains A and B, respectively. Then, the following functions can

be defined:

(f + g)(x) = f(x) + g(x) Domain: A \B.

(f � g)(x) = f(x)� g(x) Domain: A \ B.

(fg)(x) = f(x)g(x) Domain: A \B.

f(x)

g(x)Domain: x 2 A \B where g(x) 6= 0.

Example 4: Find (a) f + g, (b)f � g, (c)fg and (d)f/g and state their domains, given f(x) =p2� x

and g(x) =p5� x.

4 Spring 2017, Maya Johnson

But not

atSolve ( 2- xX5 - x)Z0

×= 2/5 < ¥334 Domain:l-I2]U[5T

Chapter 1: Sec1.3, New Functions and Old Functions

Definition: Given two functions f and g, the composite function f�g (also called the composition

of f and g) is defined by

(f � g)(x) = f(g(x))

The domain of f � g is the set of all x in the domain of g such that g(x) is in the domain of f .

Example 5: Let f(x) = x

2 and g(x) = 2x+ 3, find f � g and g � f .

Example 6: Let f(x) = 1/x and g(x) = x

2 � 1, find f � g and g � f . What is the domain in both

cases?

Example 7: If g(x) = 3x+ 2 and h(x) = 9x2 + 12x� 8, find a function f such that f � g = h.

Remark: Note that in general f � g 6= g � f , and the notation f � g means that the function g is

applied first and then f is applied second.

5 Spring 2017, Maya Johnson

( glx ) ) 2=(3×+2) (3×+2)=9×2+12×+4

Note that (glx ) ) 2- 12 = 9×2+12×+4 - 12

= 9×2+12×-8 = h ( x )

⇒ f(×)=×2Ty

Chapter 1: Sec1.3, New Functions and Old Functions

Example 8: Use the given graphs of f and g to estimate the value of each expression.

�4 �3 �2 �1 0 1 2 3 4 5 6

�4

�3

�2

�1

1

2

3

4

5

6

x

y

f(x)

g(x)

(a) f(g(2))

(b) g(f(0))

(c) (f � f)(4)

(d) (g � f)(6)

Example 9: Use the given table of f and g to evaluate the value of each expression.

x 1 2 3 4 6 8 9

f(x) 9 4 1 2 3 6 8

g(x) 4 1 8 3 2 9 6

(a) f(g(2))

(b) g(f(9))

(c) (f � f)(1)

(d) (g � f)(8)

6 Spring 2017, Maya Johnson

af ( 9)

==g918 )=D

fc , )Iqflf ' ' ' ) = fl 9) =@