Answered: flx) = In x %3D Vertical (3, In 3 =…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.5: Rational Functions

Problem 42E

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Question

The figure shows the graph of f(x) = ln x.Use transformations of this graph to graph the function g(x) = ln(x + 2). Graph and give equations of the asymptotes. Use the graphs to determine the function’s domain and range.

flx) = In x
%3D
Vertical
(3, In 3 = 1.1)
2+
asymptote:
x = 0
(2, In 2 0.7)
4
6.
10
(1,0)
,In - -0.7)
-2-
2.
1.

Expert Solution

Step 1

$T h e f i g u r e s h o w s t h e g r a p h o f f (x) = \ln x . W e h a v e t o u s e t r a n s f o r m a t i o n s o f t h i s g r a p h t o g r a p h t h e f u n c t i o n g (x) = \ln (x + 2) . T o G r a p h a n d g i v e e q u a t i o n s o f t h e a s y m p t o t e s . U s e t h e g r a p h s t o d e t e r m i n e t h e f u n c t i o n ’ s d o m a i n a n d r a n g e .$

Step 2

$As we know that {For any constant c the function f(x)=log}_{b} (x + c) s h i f t t h e f u n c t i o n l o g_{b} x l e f t i f c > 0 a n d r i g h t i f c < 0 A s h e r e c = 2 > 0, t h e f u n c t i o n g (x) = l n (x + 2) w i l l s h i f t t h e g r a p h o f f (x) = l n (x) b y 2 u n i t s l e f t . So,We can obtain the graph of ln(x+2) by horizontally shifting the graph of ln(x) by 2 units left. As we say that x=c is a vertical asymptote of the function g(x)=ln(x+2) if the limit of the function (one sided) at the point c is infinite. As \underset{x \to - 2^{+}}{l i m} l n (x + 2) = - \infty T h u s i t h a s a v e r t i c a l a s y m p t o t e x = - 2 . The graph f(x)=ln(x) has a vertical asymptote x=0 Now the asymptote of g(x)=ln(x+2) changes to x=-2.$

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