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Chapter 5: Q 75. (page 298)

In Problems 71鈥� 86, use the given functionf 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 off.

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

(f) Graph f-1.

fx=ln2x-3

Short Answer

Expert verified

Part (a) 0,鈭�.

Part (b)

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

Part (d) f-1x=12ex+3.

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=ln2x-3

02

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

fx=ln2x-3

The domain off consists of all x for which 2x>0or x>0.


Therefore, the domain of the given function f 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 of f that the range of function fx=ln2x-3is 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=ln2x-3

For f-1replace fxwithy,

role="math" localid="1650510093299" y=ln2x-3x=ln2y-3ln2y=x+32y=ex+3y=12ex+3

Therefore, the inverse of fisf-1x=12ex+3.

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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Most popular questions from this chapter

Solve given equation and verify your result using graphing utility.

(e4)x.ex2=e12

Use the graph of y=f(x)given in the Problem 43to evaluate the following:

(a) f(-1)

(b) f(1)

(c) f-1(1)

(d) f-1(2)

solve each equation. Verify your results using a graphing utility.

4x2x2=162

The function f(x)=x4is not one-to-one. Find a suitable restriction on the domain of f so that the new function that results is one-to-one. Then find the inverse of f.

Solve the equation3-x=81

and verify your results using a graphing utility.
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