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Chapter 5: Q 74. (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 f-1.

(f) Graph f-1.

fx=-ln-x

Short Answer

Expert verified

Part (a) 0,鈭�.

Part (b)

Part (c) Range -鈭�,鈭�and vertical asymptote x=0.

Part (d) f-1x=-e-x.

Part (e) Domain -鈭�,鈭�and range 0,鈭�.

Part (f)

Step by step solution

01

Part (a) Step 1. Given information.

The given function is:

fx=-ln-x

02

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

fx=-ln-x

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


Therefore, the domain of the given function is0,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 off that the range of function fx=-ln-xis the set of all real numbers.

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


The vertical asymptote of the given function isx=0.

05

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

fx=-ln-x

For f-1replace fxwithy,

y=-ln-xln-y=-x-y=e-xy=-e-x

Therefore, the inverse of fisf-1x=-e-x.

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 is0,鈭�.

07

Part (f) Graph f-1.

Sketch the graph of f-1:

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