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) =@
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