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Chapter 5: Q. 33 (page 349)

Use the given function f(x)=3ex-2to:

(a) Find the domain of f.

(b) Graph f.

(c) From the graph, determine the range and any asymptotes of f.

(d) Find f-1, the inverse of f.

(e) Find the domain and the range of f-1.

(f) Graph f-1.

Short Answer

Expert verified

Part (a) Domain of f:(-鈭�,鈭�)

Part (b) Graph of f:

Part (c) Range of f: (0,鈭�)

Horizontal asymptote:y=0

Part (d)f-1(x)=2+lnx3

Part (e) Domain of f-1:(0,鈭�)

Range off-1:(-鈭�,鈭�)

Part (f) Graph off-1:

Step by step solution

01

Part (a) Step 1. Given information 

A function,f(x)=3ex-2

02

Part (a) Step 2. Domain of the function

xcan take any real value. Therefore, domain is: (-鈭�,鈭�).

03

Part (b) Step 1. Graph of the function

Graph of the function is:

04

Part (c) Step 1. Range and asymptote

From the graph, for the given domain, range of f : (0,鈭�)

Horizontal asymptote: y=0

05

Part (d) Step 1. Inverse of function

In the function, replace x by f-1(x)and f(x)by x and rearrange the variables to get inverse of the function.

x=3e(f-1(x)-2)

x3=ef-1(x)-2

taking log to the base e on both sides

lnx3=f-1(x)-2lne

f-1(x)=2+lnx3

06

Part (e) Step 1. Domain and Range of inverse function

f-1(x)=2+lnx3

x3>0鈬�x>0

Therefore, domain of f-1:(0,鈭�)

Range off-1:(-鈭�,鈭�)

07

Part (f) Step 1. Graph of inverse function

Graph of inverse function is:

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