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Chapter 5: Q. 31 (page 349)

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

(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)=3+log2x

Part (e) Domain of localid="1646565327500" f-1:(0,鈭�)

Range oflocalid="1646565337333" f-1:(-鈭�,鈭�)

Part (f) Graph off-1:

Step by step solution

01

Part (a) Step 1. Given information 

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

02

Part (a) Step 2. Finding 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. Finding range and asymptote

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

Horizontal asymptote: y=0

05

Part (d) Step 1. Finding inverse of the function

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

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

take log to the base 2 on both sides.

log2x=(f-1(x)-3)log22

f-1(x)=3+log2x

06

Part (e) Step 1. Finding domain and range of inverse function

f-1(x)=3+log2x

xshould be greater than zero for the value of function to exist. Therefore, Domain of f-1:(0,鈭�)

For the above domain the function can take any real value. Therefore, Range of f-1:(-鈭�,鈭�)

07

Part (f) Step 1. Graph of inverse function

Graph of inverse function is:

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