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 }

Question

Want to see the full answer?

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Students have asked these similar questions

use a graphing utility to sketch the graph of the function and then use the graph to help identify or approximate the domain and range of the function. f(x)= x*sqrt(9-(x^2))

use a graphing utility to sketch the graph of the function and then use the graph to help identify or approximate the domain and range of the function. f(x)=xsqrt(9-(x^2))

Calculate a (bxc) where a = i, b = j, and c = k.

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

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

Want to see the full answer?

Check out a sample textbook solution

See solution

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

Want to see the full answer?

Check out a sample textbook solution

See solution

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

Want to see the full answer?

Check out a sample textbook solution

See solution

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 }