Whether the below function is an increasing function and find out if the inverse of the function exists. If the inverse exists, find out the inverse of the function. f ( x ) = { ( x - 1) w h e n x < 0 a n d l n x w h e n x ≥ 0 }

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Answer 1

Question

Chapter 1, Problem 58CRE

To determine

Whether the below function is an increasing function and find out if the inverse of the function exists. If the inverse exists, find out the inverse of the function.

$f (x) = {(x - 1) w h e n x < 0 a n d l n x w h e n x \geq 0}$

To determine

To find if the inverse of the function exists:

For the interval $(\infty, 0)$ , $f (x) = x - 1$

Let us assume that if $f (- a) = f (- b)$

Then, $a - 1 = b - 1$

$a = b$

Thus, the inverse of the function $f (x) = x - 1$ exists.

For the interval $[0, \infty)$ , $f (x) = l n x$

Let us assume that if $f (a) = f (b)$

Then, $l n a = l n b$

$a = b$

Thus, the inverse of the function $f (x) = l n x$ exists.

Hence, we get that the inverse of the function $f (x) = {(x - 1) w h e n x < 0 a n d l n x w h e n x \geq 0}$ exists.

Inverse of the function:

For the interval $(\infty, 0)$ , $f (x) = x - 1$

Replacing $f (x)$ with $y$ , we get

$y = x - 1$

On switching $y$ and $x$ , we get

$x = y - 1$

Solving for $y$ , we get

$x + 1 = y$

$y = x + 1$

Replacing $y$ with $f^{- 1} (x)$ , we get

$f^{- 1} (x) = x + 1$

For the interval $[0, \infty)$ , $f (x) = l n x$

Replacing $f (x)$ with $y$ , we get

$y = l n x$

On switching $y$ and $x$ , we get

$x = l n y$

Solving for $y$ , we get

$e^{x} = e^{l n y}$

$e^{x} = y$

$y = e^{x}$

Replacing $y$ with $f^{- 1} (x)$ , we get

$f^{- 1} (x) = e^{x}$

Thus, Inverse of function is $f^{- 1} (x) = {(x + 1) w h e n x < 0 a n d e^{x} w h e n x \geq 0}$

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Answer 2

Question

Chapter 1, Problem 58CRE

To determine

Whether the below function is an increasing function and find out if the inverse of the function exists. If the inverse exists, find out the inverse of the function.

$f (x) = {(x - 1) w h e n x < 0 a n d l n x w h e n x \geq 0}$

To determine

To find if the inverse of the function exists:

For the interval $(\infty, 0)$ , $f (x) = x - 1$

Let us assume that if $f (- a) = f (- b)$

Then, $a - 1 = b - 1$

$a = b$

Thus, the inverse of the function $f (x) = x - 1$ exists.

For the interval $[0, \infty)$ , $f (x) = l n x$

Let us assume that if $f (a) = f (b)$

Then, $l n a = l n b$

$a = b$

Thus, the inverse of the function $f (x) = l n x$ exists.

Hence, we get that the inverse of the function $f (x) = {(x - 1) w h e n x < 0 a n d l n x w h e n x \geq 0}$ exists.

Inverse of the function:

For the interval $(\infty, 0)$ , $f (x) = x - 1$

Replacing $f (x)$ with $y$ , we get

$y = x - 1$

On switching $y$ and $x$ , we get

$x = y - 1$

Solving for $y$ , we get

$x + 1 = y$

$y = x + 1$

Replacing $y$ with $f^{- 1} (x)$ , we get

$f^{- 1} (x) = x + 1$

For the interval $[0, \infty)$ , $f (x) = l n x$

Replacing $f (x)$ with $y$ , we get

$y = l n x$

On switching $y$ and $x$ , we get

$x = l n y$

Solving for $y$ , we get

$e^{x} = e^{l n y}$

$e^{x} = y$

$y = e^{x}$

Replacing $y$ with $f^{- 1} (x)$ , we get

$f^{- 1} (x) = e^{x}$

Thus, Inverse of function is $f^{- 1} (x) = {(x + 1) w h e n x < 0 a n d e^{x} w h e n x \geq 0}$

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Answer 3

Question

Chapter 1, Problem 58CRE

To determine

Whether the below function is an increasing function and find out if the inverse of the function exists. If the inverse exists, find out the inverse of the function.

$f (x) = {(x - 1) w h e n x < 0 a n d l n x w h e n x \geq 0}$

To determine

To find if the inverse of the function exists:

For the interval $(\infty, 0)$ , $f (x) = x - 1$

Let us assume that if $f (- a) = f (- b)$

Then, $a - 1 = b - 1$

$a = b$

Thus, the inverse of the function $f (x) = x - 1$ exists.

For the interval $[0, \infty)$ , $f (x) = l n x$

Let us assume that if $f (a) = f (b)$

Then, $l n a = l n b$

$a = b$

Thus, the inverse of the function $f (x) = l n x$ exists.

Hence, we get that the inverse of the function $f (x) = {(x - 1) w h e n x < 0 a n d l n x w h e n x \geq 0}$ exists.

Inverse of the function:

For the interval $(\infty, 0)$ , $f (x) = x - 1$

Replacing $f (x)$ with $y$ , we get

$y = x - 1$

On switching $y$ and $x$ , we get

$x = y - 1$

Solving for $y$ , we get

$x + 1 = y$

$y = x + 1$

Replacing $y$ with $f^{- 1} (x)$ , we get

$f^{- 1} (x) = x + 1$

For the interval $[0, \infty)$ , $f (x) = l n x$

Replacing $f (x)$ with $y$ , we get

$y = l n x$

On switching $y$ and $x$ , we get

$x = l n y$

Solving for $y$ , we get

$e^{x} = e^{l n y}$

$e^{x} = y$

$y = e^{x}$

Replacing $y$ with $f^{- 1} (x)$ , we get

$f^{- 1} (x) = e^{x}$

Thus, Inverse of function is $f^{- 1} (x) = {(x + 1) w h e n x < 0 a n d e^{x} w h e n x \geq 0}$

Expert Solution & Answer

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Answer 4

Question

Chapter 1, Problem 58CRE

To determine

Whether the below function is an increasing function and find out if the inverse of the function exists. If the inverse exists, find out the inverse of the function.

$f (x) = {(x - 1) w h e n x < 0 a n d l n x w h e n x \geq 0}$

To determine

To find if the inverse of the function exists:

For the interval $(\infty, 0)$ , $f (x) = x - 1$

Let us assume that if $f (- a) = f (- b)$

Then, $a - 1 = b - 1$

$a = b$

Thus, the inverse of the function $f (x) = x - 1$ exists.

For the interval $[0, \infty)$ , $f (x) = l n x$

Let us assume that if $f (a) = f (b)$

Then, $l n a = l n b$

$a = b$

Thus, the inverse of the function $f (x) = l n x$ exists.

Hence, we get that the inverse of the function $f (x) = {(x - 1) w h e n x < 0 a n d l n x w h e n x \geq 0}$ exists.

Inverse of the function:

For the interval $(\infty, 0)$ , $f (x) = x - 1$

Replacing $f (x)$ with $y$ , we get

$y = x - 1$

On switching $y$ and $x$ , we get

$x = y - 1$

Solving for $y$ , we get

$x + 1 = y$

$y = x + 1$

Replacing $y$ with $f^{- 1} (x)$ , we get

$f^{- 1} (x) = x + 1$

For the interval $[0, \infty)$ , $f (x) = l n x$

Replacing $f (x)$ with $y$ , we get

$y = l n x$

On switching $y$ and $x$ , we get

$x = l n y$

Solving for $y$ , we get

$e^{x} = e^{l n y}$

$e^{x} = y$

$y = e^{x}$

Replacing $y$ with $f^{- 1} (x)$ , we get

$f^{- 1} (x) = e^{x}$

Thus, Inverse of function is $f^{- 1} (x) = {(x + 1) w h e n x < 0 a n d e^{x} w h e n x \geq 0}$

Expert Solution & Answer

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Answer 5

Chapter 1, Problem 58CRE

To determine

Whether the below function is an increasing function and find out if the inverse of the function exists. If the inverse exists, find out the inverse of the function.

$f (x) = {(x - 1) w h e n x < 0 a n d l n x w h e n x \geq 0}$

To determine

To find if the inverse of the function exists:

For the interval $(\infty, 0)$ , $f (x) = x - 1$

Let us assume that if $f (- a) = f (- b)$

Then, $a - 1 = b - 1$

$a = b$

Thus, the inverse of the function $f (x) = x - 1$ exists.

For the interval $[0, \infty)$ , $f (x) = l n x$

Let us assume that if $f (a) = f (b)$

Then, $l n a = l n b$

$a = b$

Thus, the inverse of the function $f (x) = l n x$ exists.

Hence, we get that the inverse of the function $f (x) = {(x - 1) w h e n x < 0 a n d l n x w h e n x \geq 0}$ exists.

Inverse of the function:

For the interval $(\infty, 0)$ , $f (x) = x - 1$

Replacing $f (x)$ with $y$ , we get

$y = x - 1$

On switching $y$ and $x$ , we get

$x = y - 1$

Solving for $y$ , we get

$x + 1 = y$

$y = x + 1$

Replacing $y$ with $f^{- 1} (x)$ , we get

$f^{- 1} (x) = x + 1$

For the interval $[0, \infty)$ , $f (x) = l n x$

Replacing $f (x)$ with $y$ , we get

$y = l n x$

On switching $y$ and $x$ , we get

$x = l n y$

Solving for $y$ , we get

$e^{x} = e^{l n y}$

$e^{x} = y$

$y = e^{x}$

Replacing $y$ with $f^{- 1} (x)$ , we get

$f^{- 1} (x) = e^{x}$

Thus, Inverse of function is $f^{- 1} (x) = {(x + 1) w h e n x < 0 a n d e^{x} w h e n x \geq 0}$

Answer 6

Chapter 1, Problem 58CRE

To determine

Whether the below function is an increasing function and find out if the inverse of the function exists. If the inverse exists, find out the inverse of the function.

$f (x) = {(x - 1) w h e n x < 0 a n d l n x w h e n x \geq 0}$

Answer 7

Chapter 1, Problem 58CRE

To determine

Whether the below function is an increasing function and find out if the inverse of the function exists. If the inverse exists, find out the inverse of the function.

$f (x) = {(x - 1) w h e n x < 0 a n d l n x w h e n x \geq 0}$

Answer 8

To determine

To find if the inverse of the function exists:

For the interval $(\infty, 0)$ , $f (x) = x - 1$

Let us assume that if $f (- a) = f (- b)$

Then, $a - 1 = b - 1$

$a = b$

Thus, the inverse of the function $f (x) = x - 1$ exists.

For the interval $[0, \infty)$ , $f (x) = l n x$

Let us assume that if $f (a) = f (b)$

Then, $l n a = l n b$

$a = b$

Thus, the inverse of the function $f (x) = l n x$ exists.

Hence, we get that the inverse of the function $f (x) = {(x - 1) w h e n x < 0 a n d l n x w h e n x \geq 0}$ exists.

Inverse of the function:

For the interval $(\infty, 0)$ , $f (x) = x - 1$

Replacing $f (x)$ with $y$ , we get

$y = x - 1$

On switching $y$ and $x$ , we get

$x = y - 1$

Solving for $y$ , we get

$x + 1 = y$

$y = x + 1$

Replacing $y$ with $f^{- 1} (x)$ , we get

$f^{- 1} (x) = x + 1$

For the interval $[0, \infty)$ , $f (x) = l n x$

Replacing $f (x)$ with $y$ , we get

$y = l n x$

On switching $y$ and $x$ , we get

$x = l n y$

Solving for $y$ , we get

$e^{x} = e^{l n y}$

$e^{x} = y$

$y = e^{x}$

Replacing $y$ with $f^{- 1} (x)$ , we get

$f^{- 1} (x) = e^{x}$

Thus, Inverse of function is $f^{- 1} (x) = {(x + 1) w h e n x < 0 a n d e^{x} w h e n x \geq 0}$

Answer 9

To determine

To find if the inverse of the function exists:

For the interval $(\infty, 0)$ , $f (x) = x - 1$

Let us assume that if $f (- a) = f (- b)$

Then, $a - 1 = b - 1$

$a = b$

Thus, the inverse of the function $f (x) = x - 1$ exists.

For the interval $[0, \infty)$ , $f (x) = l n x$

Let us assume that if $f (a) = f (b)$

Then, $l n a = l n b$

$a = b$

Thus, the inverse of the function $f (x) = l n x$ exists.

Hence, we get that the inverse of the function $f (x) = {(x - 1) w h e n x < 0 a n d l n x w h e n x \geq 0}$ exists.

Inverse of the function:

For the interval $(\infty, 0)$ , $f (x) = x - 1$

Replacing $f (x)$ with $y$ , we get

$y = x - 1$

On switching $y$ and $x$ , we get

$x = y - 1$

Solving for $y$ , we get

$x + 1 = y$

$y = x + 1$

Replacing $y$ with $f^{- 1} (x)$ , we get

$f^{- 1} (x) = x + 1$

For the interval $[0, \infty)$ , $f (x) = l n x$

Replacing $f (x)$ with $y$ , we get

$y = l n x$

On switching $y$ and $x$ , we get

$x = l n y$

Solving for $y$ , we get

$e^{x} = e^{l n y}$

$e^{x} = y$

$y = e^{x}$

Replacing $y$ with $f^{- 1} (x)$ , we get

$f^{- 1} (x) = e^{x}$

Thus, Inverse of function is $f^{- 1} (x) = {(x + 1) w h e n x < 0 a n d e^{x} w h e n x \geq 0}$

Answer 10

Expert Solution & Answer

Answer 11

Expert Solution & Answer

Answer 12

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Answer 13

Ch. 1.1 - Prob. 1PQ Ch. 1.1 - Prob. 2PQ Ch. 1.1 - Prob. 3PQ Ch. 1.1 - Prob. 4PQ Ch. 1.1 - Prob. 5PQ Ch. 1.1 - Prob. 6PQ Ch. 1.1 - Prob. 7PQ Ch. 1.1 - Prob. 8PQ Ch. 1.1 - Prob. 9PQ Ch. 1.1 - Prob. 10PQ

Whether the below function is an increasing function and find out if the inverse of the function exists. If the inverse exists, find out the inverse of the function. f ( x ) = { ( x - 1) w h e n x &lt; 0 a n d l n x w h e n x ≥ 0 }