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Chapter 5: Q. 13 (page 351)

In Problems 12 and 13, use the given function f to:

(a) Find the domain off.

(b) Graphf.

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

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

(e) Find the domain and the range off-1.

(f) Graph f-1

f(x)=1-log5(x-2)

Short Answer

Expert verified

(a) The domain is2,鈭�

(b) The graph of the function is shown in step 3.

(c) Range is -鈭�,鈭�and vertical asymptotes is x=2.

(d) The inverse of f is f-1x=51-x+2.

(e) Domain of f-1is -鈭�,鈭�and range is2,鈭�

(f) The graph of inverse function is shown in step 9.

Step by step solution

01

Step 1. Given Information

The given function isfx=1-log5x-2

02

Part (a) Step 1. Finding domain of the function

fx=1-log5x-2

thenx-2>0x>2

03

Part (b) . Step 1. Graph of the function f

The graph is shown in the figure

04

Part (c) Step 1. Range of the function f

From the graph we see that the range of the function is the set of real numbers-鈭�,鈭�

05

Part (c) Step 2. Asymptote of the function 

From the graph we see that the function has a vertical asymptote is x=2because the function approaches the line x=2and does not intersect it.

06

Part (d). Step 1. Finding inverse of f

Put fx=y

1-log5x-2=y

log5y-2=1-xy-2=51-xy=2+51-xf-1x=51-x+2

07

Part (e). Step 1. Domain and range of f-1

Range of f-1=2,鈭�

Domain off-1=-鈭�,鈭�

08

Part (f) Step 1. Graph of f-1.

The graph of inverse function is shown in the figure

09

Part (f) 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.

f(x)=2x-3x+4

In Problems 41-52, use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function.

y=2x+1

A function y=f(x)is decreasing on the interval(0,5).What conclusions can you draw about the graph ofy=f-1(x)?

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

True or False

The function f(x)=2xx-3 hasy=2 as a horizontal asymptote.

Solve the equation3-x=81

and verify your results using a graphing utility.
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