The inverse of f and g , and use these to find when f ( x ) = 2 and g ( x ) = 2 .

Question

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Answer 1

Question

Chapter 1, Problem 3SP

a)

To determine

The inverse of f and g , and use these to find when f(x)=2 and g(x)=2 .

a)

Expert Solution

Answer to Problem 3SP

f−1(x)=−ln(x)2g−1(x)=(x−1)13

Explanation of Solution

Given:

Given the functions

f(x)=e−2xg(x)=x3+1

Calculation:

First find the inverse of f(x)

Let f(x)=y

So,

y=e−2x

Now interchange the variable x and y

x=e−2ytaking ln both sidesln(x)=−2yy=−ln(x)2

Therefore, the inverse of f(x)

f−1(x)=−ln(x)2

Now find the inverse of g(x)

g(x)=x3+1let y=g(x)so,y=x3+1x3=y−1x=(y−1)13

Therefore, the inverse of g(x)

g−1(x)=(x−1)13

By using of f and g

f(x)=2e−2x=2taking ln both sides−2x=ln(2)x=−ln(2)2

And

g(x)=2x3+1=2x3=1x=1

b)

To determine

The fοg and gοf , and evaluate each at x=2 .

b)

Expert Solution

Answer to Problem 3SP

fοg=e−2(x2+1)fοg(2)=1.523×10−8gοf=e−6x+1gοf=1.0000

Explanation of Solution

Given:

Given the functions

f(x)=e−2xg(x)=x3+1

Calculation:

First find fοg

fοg=e−2(x3+1)

At x=2

fοg=e−2(8+1)fοg=e−18fοg=1.523×10−8

Now find gοfgοf=(e−2x)3+1gοf=e−6x+1

At x=2

gοf=e−6(2)+1gοf=e−12+1gοf=1.0000

c)

To determine

The inverse of gοf , and the domain of the new function.

c)

Expert Solution

Answer to Problem 3SP

gοf−1(x)=−16ln(x−1)

Domain- (1,∞)

Explanation of Solution

Given:

Given the functions

gοf=e−6x+1

Calculation:

let gοf=yy=e−6x+1e−6x=y−1taking ln both sides−6x=ln(y−1)x=−16ln(y−1)so,gοf−1(x)=−16ln(x−1)

Now find the domain of the new function

x−1>0x>1

Therefore. The domain of the function

x>1(1,∞)

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Answer 2

Question

Chapter 1, Problem 3SP

a)

To determine

The inverse of f and g , and use these to find when f(x)=2 and g(x)=2 .

a)

Expert Solution

Answer to Problem 3SP

f−1(x)=−ln(x)2g−1(x)=(x−1)13

Explanation of Solution

Given:

Given the functions

f(x)=e−2xg(x)=x3+1

Calculation:

First find the inverse of f(x)

Let f(x)=y

So,

y=e−2x

Now interchange the variable x and y

x=e−2ytaking ln both sidesln(x)=−2yy=−ln(x)2

Therefore, the inverse of f(x)

f−1(x)=−ln(x)2

Now find the inverse of g(x)

g(x)=x3+1let y=g(x)so,y=x3+1x3=y−1x=(y−1)13

Therefore, the inverse of g(x)

g−1(x)=(x−1)13

By using of f and g

f(x)=2e−2x=2taking ln both sides−2x=ln(2)x=−ln(2)2

And

g(x)=2x3+1=2x3=1x=1

b)

To determine

The fοg and gοf , and evaluate each at x=2 .

b)

Expert Solution

Answer to Problem 3SP

fοg=e−2(x2+1)fοg(2)=1.523×10−8gοf=e−6x+1gοf=1.0000

Explanation of Solution

Given:

Given the functions

f(x)=e−2xg(x)=x3+1

Calculation:

First find fοg

fοg=e−2(x3+1)

At x=2

fοg=e−2(8+1)fοg=e−18fοg=1.523×10−8

Now find gοfgοf=(e−2x)3+1gοf=e−6x+1

At x=2

gοf=e−6(2)+1gοf=e−12+1gοf=1.0000

c)

To determine

The inverse of gοf , and the domain of the new function.

c)

Expert Solution

Answer to Problem 3SP

gοf−1(x)=−16ln(x−1)

Domain- (1,∞)

Explanation of Solution

Given:

Given the functions

gοf=e−6x+1

Calculation:

let gοf=yy=e−6x+1e−6x=y−1taking ln both sides−6x=ln(y−1)x=−16ln(y−1)so,gοf−1(x)=−16ln(x−1)

Now find the domain of the new function

x−1>0x>1

Therefore. The domain of the function

x>1(1,∞)

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Answer 3

Question

Chapter 1, Problem 3SP

a)

To determine

The inverse of f and g , and use these to find when f(x)=2 and g(x)=2 .

a)

Expert Solution

Answer to Problem 3SP

f−1(x)=−ln(x)2g−1(x)=(x−1)13

Explanation of Solution

Given:

Given the functions

f(x)=e−2xg(x)=x3+1

Calculation:

First find the inverse of f(x)

Let f(x)=y

So,

y=e−2x

Now interchange the variable x and y

x=e−2ytaking ln both sidesln(x)=−2yy=−ln(x)2

Therefore, the inverse of f(x)

f−1(x)=−ln(x)2

Now find the inverse of g(x)

g(x)=x3+1let y=g(x)so,y=x3+1x3=y−1x=(y−1)13

Therefore, the inverse of g(x)

g−1(x)=(x−1)13

By using of f and g

f(x)=2e−2x=2taking ln both sides−2x=ln(2)x=−ln(2)2

And

g(x)=2x3+1=2x3=1x=1

b)

To determine

The fοg and gοf , and evaluate each at x=2 .

b)

Expert Solution

Answer to Problem 3SP

fοg=e−2(x2+1)fοg(2)=1.523×10−8gοf=e−6x+1gοf=1.0000

Explanation of Solution

Given:

Given the functions

f(x)=e−2xg(x)=x3+1

Calculation:

First find fοg

fοg=e−2(x3+1)

At x=2

fοg=e−2(8+1)fοg=e−18fοg=1.523×10−8

Now find gοfgοf=(e−2x)3+1gοf=e−6x+1

At x=2

gοf=e−6(2)+1gοf=e−12+1gοf=1.0000

c)

To determine

The inverse of gοf , and the domain of the new function.

c)

Expert Solution

Answer to Problem 3SP

gοf−1(x)=−16ln(x−1)

Domain- (1,∞)

Explanation of Solution

Given:

Given the functions

gοf=e−6x+1

Calculation:

let gοf=yy=e−6x+1e−6x=y−1taking ln both sides−6x=ln(y−1)x=−16ln(y−1)so,gοf−1(x)=−16ln(x−1)

Now find the domain of the new function

x−1>0x>1

Therefore. The domain of the function

x>1(1,∞)

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Answer 4

Question

Chapter 1, Problem 3SP

a)

To determine

The inverse of f and g , and use these to find when f(x)=2 and g(x)=2 .

a)

Expert Solution

Answer to Problem 3SP

f−1(x)=−ln(x)2g−1(x)=(x−1)13

Explanation of Solution

Given:

Given the functions

f(x)=e−2xg(x)=x3+1

Calculation:

First find the inverse of f(x)

Let f(x)=y

So,

y=e−2x

Now interchange the variable x and y

x=e−2ytaking ln both sidesln(x)=−2yy=−ln(x)2

Therefore, the inverse of f(x)

f−1(x)=−ln(x)2

Now find the inverse of g(x)

g(x)=x3+1let y=g(x)so,y=x3+1x3=y−1x=(y−1)13

Therefore, the inverse of g(x)

g−1(x)=(x−1)13

By using of f and g

f(x)=2e−2x=2taking ln both sides−2x=ln(2)x=−ln(2)2

And

g(x)=2x3+1=2x3=1x=1

b)

To determine

The fοg and gοf , and evaluate each at x=2 .

b)

Expert Solution

Answer to Problem 3SP

fοg=e−2(x2+1)fοg(2)=1.523×10−8gοf=e−6x+1gοf=1.0000

Explanation of Solution

Given:

Given the functions

f(x)=e−2xg(x)=x3+1

Calculation:

First find fοg

fοg=e−2(x3+1)

At x=2

fοg=e−2(8+1)fοg=e−18fοg=1.523×10−8

Now find gοfgοf=(e−2x)3+1gοf=e−6x+1

At x=2

gοf=e−6(2)+1gοf=e−12+1gοf=1.0000

c)

To determine

The inverse of gοf , and the domain of the new function.

c)

Expert Solution

Answer to Problem 3SP

gοf−1(x)=−16ln(x−1)

Domain- (1,∞)

Explanation of Solution

Given:

Given the functions

gοf=e−6x+1

Calculation:

let gοf=yy=e−6x+1e−6x=y−1taking ln both sides−6x=ln(y−1)x=−16ln(y−1)so,gοf−1(x)=−16ln(x−1)

Now find the domain of the new function

x−1>0x>1

Therefore. The domain of the function

x>1(1,∞)

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Answer 5

Chapter 1, Problem 3SP

a)

To determine

The inverse of f and g , and use these to find when f(x)=2 and g(x)=2 .

a)

Expert Solution

Answer to Problem 3SP

f−1(x)=−ln(x)2g−1(x)=(x−1)13

Explanation of Solution

Given:

Given the functions

f(x)=e−2xg(x)=x3+1

Calculation:

First find the inverse of f(x)

Let f(x)=y

So,

y=e−2x

Now interchange the variable x and y

x=e−2ytaking ln both sidesln(x)=−2yy=−ln(x)2

Therefore, the inverse of f(x)

f−1(x)=−ln(x)2

Now find the inverse of g(x)

g(x)=x3+1let y=g(x)so,y=x3+1x3=y−1x=(y−1)13

Therefore, the inverse of g(x)

g−1(x)=(x−1)13

By using of f and g

f(x)=2e−2x=2taking ln both sides−2x=ln(2)x=−ln(2)2

And

g(x)=2x3+1=2x3=1x=1

b)

To determine

The fοg and gοf , and evaluate each at x=2 .

b)

Expert Solution

Answer to Problem 3SP

fοg=e−2(x2+1)fοg(2)=1.523×10−8gοf=e−6x+1gοf=1.0000

Explanation of Solution

Given:

Given the functions

f(x)=e−2xg(x)=x3+1

Calculation:

First find fοg

fοg=e−2(x3+1)

At x=2

fοg=e−2(8+1)fοg=e−18fοg=1.523×10−8

Now find gοfgοf=(e−2x)3+1gοf=e−6x+1

At x=2

gοf=e−6(2)+1gοf=e−12+1gοf=1.0000

c)

To determine

The inverse of gοf , and the domain of the new function.

c)

Expert Solution

Answer to Problem 3SP

gοf−1(x)=−16ln(x−1)

Domain- (1,∞)

Explanation of Solution

Given:

Given the functions

gοf=e−6x+1

Calculation:

let gοf=yy=e−6x+1e−6x=y−1taking ln both sides−6x=ln(y−1)x=−16ln(y−1)so,gοf−1(x)=−16ln(x−1)

Now find the domain of the new function

x−1>0x>1

Therefore. The domain of the function

x>1(1,∞)

Answer 6

Chapter 1, Problem 3SP

a)

To determine

The inverse of f and g , and use these to find when f(x)=2 and g(x)=2 .

a)

Expert Solution

Answer to Problem 3SP

f−1(x)=−ln(x)2g−1(x)=(x−1)13

Explanation of Solution

Given:

Given the functions

f(x)=e−2xg(x)=x3+1

Calculation:

First find the inverse of f(x)

Let f(x)=y

So,

y=e−2x

Now interchange the variable x and y

x=e−2ytaking ln both sidesln(x)=−2yy=−ln(x)2

Therefore, the inverse of f(x)

f−1(x)=−ln(x)2

Now find the inverse of g(x)

g(x)=x3+1let y=g(x)so,y=x3+1x3=y−1x=(y−1)13

Therefore, the inverse of g(x)

g−1(x)=(x−1)13

By using of f and g

f(x)=2e−2x=2taking ln both sides−2x=ln(2)x=−ln(2)2

And

g(x)=2x3+1=2x3=1x=1

Answer 7

Chapter 1, Problem 3SP

a)

To determine

The inverse of f and g , and use these to find when f(x)=2 and g(x)=2 .

Answer 8

a)

Expert Solution

Answer to Problem 3SP

f−1(x)=−ln(x)2g−1(x)=(x−1)13

Answer 9

a)

Expert Solution

Answer 10

a)

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

Given:

Given the functions

f(x)=e−2xg(x)=x3+1

Calculation:

First find the inverse of f(x)

Let f(x)=y

So,

y=e−2x

Now interchange the variable x and y

x=e−2ytaking ln both sidesln(x)=−2yy=−ln(x)2

Therefore, the inverse of f(x)

f−1(x)=−ln(x)2

Now find the inverse of g(x)

g(x)=x3+1let y=g(x)so,y=x3+1x3=y−1x=(y−1)13

Therefore, the inverse of g(x)

g−1(x)=(x−1)13

By using of f and g

f(x)=2e−2x=2taking ln both sides−2x=ln(2)x=−ln(2)2

And

g(x)=2x3+1=2x3=1x=1

Answer 13

b)

To determine

The fοg and gοf , and evaluate each at x=2 .

b)

Expert Solution

Answer to Problem 3SP

fοg=e−2(x2+1)fοg(2)=1.523×10−8gοf=e−6x+1gοf=1.0000

Explanation of Solution

Given:

Given the functions

f(x)=e−2xg(x)=x3+1

Calculation:

First find fοg

fοg=e−2(x3+1)

At x=2

fοg=e−2(8+1)fοg=e−18fοg=1.523×10−8

Now find gοfgοf=(e−2x)3+1gοf=e−6x+1

At x=2

gοf=e−6(2)+1gοf=e−12+1gοf=1.0000

Answer 14

b)

To determine

The fοg and gοf , and evaluate each at x=2 .

Answer 15

b)

Expert Solution

Answer to Problem 3SP

fοg=e−2(x2+1)fοg(2)=1.523×10−8gοf=e−6x+1gοf=1.0000

Answer 16

b)

Expert Solution

Answer 17

b)

Expert Solution

Answer 18

Expert Solution

Answer 19

Explanation of Solution

Given:

Given the functions

f(x)=e−2xg(x)=x3+1

Calculation:

First find fοg

fοg=e−2(x3+1)

At x=2

fοg=e−2(8+1)fοg=e−18fοg=1.523×10−8

Now find gοfgοf=(e−2x)3+1gοf=e−6x+1

At x=2

gοf=e−6(2)+1gοf=e−12+1gοf=1.0000

Answer 20

c)

To determine

The inverse of gοf , and the domain of the new function.

c)

Expert Solution

Answer to Problem 3SP

gοf−1(x)=−16ln(x−1)

Domain- (1,∞)

Explanation of Solution

Given:

Given the functions

gοf=e−6x+1

Calculation:

let gοf=yy=e−6x+1e−6x=y−1taking ln both sides−6x=ln(y−1)x=−16ln(y−1)so,gοf−1(x)=−16ln(x−1)

Now find the domain of the new function

x−1>0x>1

Therefore. The domain of the function

x>1(1,∞)

Answer 21

c)

To determine

The inverse of gοf , and the domain of the new function.

Answer 22

c)

Expert Solution

Answer to Problem 3SP

gοf−1(x)=−16ln(x−1)

Domain- (1,∞)

Answer 23

c)

Expert Solution

Answer 24

c)

Expert Solution

Answer 25

Expert Solution

Answer 26

Explanation of Solution

Given:

Given the functions

gοf=e−6x+1

Calculation:

let gοf=yy=e−6x+1e−6x=y−1taking ln both sides−6x=ln(y−1)x=−16ln(y−1)so,gοf−1(x)=−16ln(x−1)

Now find the domain of the new function

x−1>0x>1

Therefore. The domain of the function

x>1(1,∞)

Answer 27

Ch. 1.2 - Identify the variables and parameters in the...Ch. 1.2 - Prob. 2E Ch. 1.2 - Compute the values of the following functions at...Ch. 1.2 - Compute the values of the following functions at...Ch. 1.2 - Compute the values of the following functions at...Ch. 1.2 - Compute the values of the following functions at...Ch. 1.2 - Graph the given points and say which point does...Ch. 1.2 - Graph the given points and say which point does...Ch. 1.2 - Graph the given points and say which point does...Ch. 1.2 - Graph the given points and say which point does...

The inverse of f and g , and use these to find when f ( x ) = 2 and g ( x ) = 2 .

Videos

The inverse of f and g , and use these to find when f(x)=2 and g(x)=2 .

Answer to Problem 3SP

Explanation of Solution

The fοg and gοf , and evaluate each at x=2 .

Answer to Problem 3SP

Explanation of Solution

The inverse of gοf , and the domain of the new function.

Answer to Problem 3SP

Explanation of Solution

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Chapter 1 Solutions