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Chapter 5: Q 77. (page 298)

In Problems 71鈥� 86, use the given functionf to:

(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 localid="1650525373122" f-1.

(f) Graph localid="1650525363664" f-1.

localid="1650525345524" fx=logx-4+2

Short Answer

Expert verified

Part (a) 4,鈭�.

Part (b)

Part (c) Range -鈭�,鈭�and vertical asymptote x=4.

Part (d)f-1x=10x-2+4.

Part (e) Domain -鈭�,鈭�and range 4,鈭�.

Part (f)

Step by step solution

01

Part (a) Step 1. Given information.

The given function is:

fx=logx-4+2

02

Part (a) Step 2. Find the domain of f.

fx=logx-4+2

The domain off consists of all x for which x-4>0or x>4.


Therefore, the domain of the given function is4,鈭�.

03

Part (b) Step 1. Graph f.

Sketch the graph off:

04

Part (c) Step 1. Determine the range and any asymptotes of f  from the graph.

We can see from the graph of f that the range of function fx=logx-4+2is the set of all real numbers.

Therefore, the range of the function is -鈭�,鈭�.


The vertical asymptote of the given function isx=4.

05

Part (d) Step 1. Find f-1.

fx=logx-4+2

For f-1replace fxwithy,

y=logx-4+2x=logy-4+2logy-4=x-2y-4=10x-2y=10x-2+4

Therefore, the inverse of fisf-1x=10x-2+4.

06

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

We know that the domain of a functionf(x)is the range of its inverse.

In the same way, the range of a function f(x) is the domain of its inverse.

Therefore, the domain of f-1is-鈭�鈭�and its range is4,鈭�.

07

Part (f) Graph f-1.

Sketch the graph of f-1:

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Most popular questions from this chapter

In Problems 61鈥�72, the function f is one-to-one. Find its inverse and check your answer.

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Solve the equation. Verify your results using a graphing utility.

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