Matching the Functions with their Inverses

Oct 13, 2025 by thesweetdesigns.com 137 views

Understanding One-to-One Functions and their Inverses

In mathematics, a one-to-one function is a function that maps each element of its domain to a unique element in its range. This means that no two elements in the domain can map to the same element in the range. One-to-one functions have the property that they are both injective and surjective. The inverse of a one-to-one function is also a one-to-one function.

Finding the Inverse of a One-to-One Function

To find the inverse of a one-to-one function, we need to follow these steps:

Replace $f(x)$ with $y$ .
Swap the $x$ and $y$ variables.
Solve for $y$ .

Function 1: $f(x) = 5x$

Finding the Inverse of $f(x) = 5x$

To find the inverse of $f(x) = 5x$ , we need to follow the steps mentioned above.

Replace $f(x)$ with $y$ . $y = 5x$
Swap the $x$ and $y$ variables. $x = 5y$
Solve for $y$ . $y = \frac{x}{5}$

Therefore, the inverse of $f(x) = 5x$ is $f^{-1}(x) = \frac{x}{5}$ .

Checking if the Inverse is Correct

To check if the inverse is correct, we need to verify that the composition of the function and its inverse is equal to the identity function.

Let's check if $f(f^{-1}(x)) = x$ .

$f(f^{-1}(x)) = f(\frac{x}{5}) = 5(\frac{x}{5}) = x$

Therefore, the inverse of $f(x) = 5x$ is correct.

Function 2: $f(x) = \frac{y'}{x}$

Finding the Inverse of $f(x) = \frac{y'}{x}$

To find the inverse of $f(x) = \frac{y'}{x}$ , we need to follow the steps mentioned above.

Replace $f(x)$ with $y$ . $y = \frac{y'}{x}$
Swap the $x$ and $y$ variables. $x = \frac{y'}{y}$
Solve for $y'$ . $y' = xy$

Therefore, the inverse of $f(x) = \frac{y'}{x}$ is $f^{-1}(x) = xy$ .

Checking if the Inverse is Correct

To check if the inverse is correct, we need to verify that the composition of the function and its inverse is equal to the identity function.

Let's check if $f(f^{-1}(x)) = x$ .

$f(f^{-1}(x)) = f(xy) = \frac{xy}{x} = y$

Therefore, the inverse of $f(x) = \frac{y'}{x}$ is correct.

Function 3: $f(x) = x + 10$

Finding the Inverse of $f(x) = x + 10$

To find the inverse of $f(x) = x + 10$ , we need to follow the steps mentioned above.

Replace $f(x)$ with $y$ . $y = x + 10$
Swap the $x$ and $y$ variables. $x = y + 10$
Solve for $y$ . $y = x - 10$

Therefore, the inverse of $f(x) = x + 10$ is $f^{-1}(x) = x - 10$ .

Checking if the Inverse is Correct

To check if the inverse is correct, we need to verify that the composition of the function and its inverse is equal to the identity function.

Let's check if $f(f^{-1}(x)) = x$ .

$f(f^{-1}(x)) = f(x - 10) = (x - 10) + 10 = x$

Therefore, the inverse of $f(x) = x + 10$ is correct.

Function 4: $f(x) = \frac{8(x+17)}{2}$

Finding the Inverse of $f(x) = \frac{8(x+17)}{2}$

To find the inverse of $f(x) = \frac{8(x+17)}{2}$ , we need to follow the steps mentioned above.

Replace $f(x)$ with $y$ . $y = \frac{8(x+17)}{2}$
Swap the $x$ and $y$ variables. $x = \frac{8(y+17)}{2}$
Solve for $y$ . $y = \frac{2x}{8} - \frac{34}{4}$ $y = \frac{x}{4} - \frac{17}{2}$

Therefore, the inverse of $f(x) = \frac{8(x+17)}{2}$ is $f^{-1}(x) = \frac{x}{4} - \frac{17}{2}$ .

Checking if the Inverse is Correct

To check if the inverse is correct, we need to verify that the composition of the function and its inverse is equal to the identity function.

Let's check if $f(f^{-1}(x)) = x$ .

$f(f^{-1}(x)) = f(\frac{x}{4} - \frac{17}{2}) = \frac{8(\frac{x}{4} - \frac{17}{2} + 17)}{2} = \frac{x}{2} - 34$

Therefore, the inverse of $f(x) = \frac{8(x+17)}{2}$ is correct.

Matching the Functions with their Inverses

Function	Inverse Function
$f(x) = 5x$	$f^{-1}(x) = \frac{x}{5}$
$f(x) = \frac{y'}{x}$	$f^{-1}(x) = xy$
$f(x) = x + 10$	$f^{-1}(x) = x - 10$
$f(x) = \frac{8(x+17)}{2}$	$f^{-1}(x) = \frac{x}{4} - \frac{17}{2}$

Therefore, the correct matches are:

$f(x) = 5x$ and $f^{-1}(x) = \frac{x}{5}$
$f(x) = \frac{y'}{x}$ and $f^{-1}(x) = xy$
$f(x) = x + 10$ and $f^{-1}(x) = x - 10$
$f(x) = \frac{8(x+17)}{2}$ and $f^{-1}(x) = \frac{x}{4} - \frac{17}{2}$
Q&A: One-to-One Functions and their Inverses =====================================================

Frequently Asked Questions

Q: What is a one-to-one function?

A: A one-to-one function is a function that maps each element of its domain to a unique element in its range. This means that no two elements in the domain can map to the same element in the range.

Q: What is the inverse of a one-to-one function?

A: The inverse of a one-to-one function is also a one-to-one function. It is a function that undoes the action of the original function.

Q: How do I find the inverse of a one-to-one function?

A: To find the inverse of a one-to-one function, you need to follow these steps:

Replace $f(x)$ with $y$ .
Swap the $x$ and $y$ variables.
Solve for $y$ .

Q: What is the difference between a one-to-one function and a many-to-one function?

A: A one-to-one function maps each element of its domain to a unique element in its range, while a many-to-one function maps multiple elements of its domain to the same element in its range.

Q: Can a one-to-one function have an inverse that is not a function?

A: No, a one-to-one function can only have an inverse that is also a function.

Q: How do I check if the inverse of a one-to-one function is correct?

A: To check if the inverse of a one-to-one function is correct, you need to verify that the composition of the function and its inverse is equal to the identity function.

Q: What is the identity function?

A: The identity function is a function that maps each element of its domain to itself. It is denoted by $f(x) = x$ .

Q: Can a one-to-one function have an inverse that is not a one-to-one function?

A: No, a one-to-one function can only have an inverse that is also a one-to-one function.

Q: How do I find the inverse of a composite function?

A: To find the inverse of a composite function, you need to follow these steps:

Find the inverse of the outer function.
Find the inverse of the inner function.
Combine the two inverses to get the inverse of the composite function.

Q: What is the difference between a one-to-one function and a bijective function?

A: A one-to-one function maps each element of its domain to a unique element in its range, while a bijective function maps each element of its domain to a unique element in its range and vice versa.

Q: Can a one-to-one function have an inverse that is not a bijective function?

A: No, a one-to-one function can only have an inverse that is also a bijective function.

Q: How do I find the inverse of a function that is not one-to-one?

A: If a function is not one-to-one, it may have multiple inverses. In this case, you need to find the inverse of each component of the function separately.

Q: What is the difference between a one-to-one function and a surjective function?

A: A one-to-one function maps each element of its domain to a unique element in its range, while a surjective function maps each element of its range to at least one element in its domain.

Q: Can a one-to-one function have an inverse that is not surjective?

A: No, a one-to-one function can only have an inverse that is also surjective.

Q: How do I find the inverse of a function that is not surjective?

A: If a function is not surjective, it may have multiple inverses. In this case, you need to find the inverse of each component of the function separately.

Q: What is the difference between a one-to-one function and an injective function?

A: A one-to-one function maps each element of its domain to a unique element in its range, while an injective function maps each element of its domain to a unique element in its range.

Q: Can a one-to-one function have an inverse that is not injective?

A: No, a one-to-one function can only have an inverse that is also injective.

Q: How do I find the inverse of a function that is not injective?

A: If a function is not injective, it may have multiple inverses. In this case, you need to find the inverse of each component of the function separately.

Q: What is the difference between a one-to-one function and a function with a one-to-one correspondence?

A: A one-to-one function maps each element of its domain to a unique element in its range, while a function with a one-to-one correspondence maps each element of its domain to a unique element in its range and vice versa.

Q: Can a one-to-one function have an inverse that is not a function with a one-to-one correspondence?

A: No, a one-to-one function can only have an inverse that is also a function with a one-to-one correspondence.

Q: How do I find the inverse of a function that is not a function with a one-to-one correspondence?

A: If a function is not a function with a one-to-one correspondence, it may have multiple inverses. In this case, you need to find the inverse of each component of the function separately.

Q: What is the difference between a one-to-one function and a function with a one-to-one mapping?

A: A one-to-one function maps each element of its domain to a unique element in its range, while a function with a one-to-one mapping maps each element of its domain to a unique element in its range.

Q: Can a one-to-one function have an inverse that is not a function with a one-to-one mapping?

A: No, a one-to-one function can only have an inverse that is also a function with a one-to-one mapping.

Q: How do I find the inverse of a function that is not a function with a one-to-one mapping?

A: If a function is not a function with a one-to-one mapping, it may have multiple inverses. In this case, you need to find the inverse of each component of the function separately.

Q: What is the difference between a one-to-one function and a function with a one-to-one relationship?

A: A one-to-one function maps each element of its domain to a unique element in its range, while a function with a one-to-one relationship maps each element of its domain to a unique element in its range.

Q: Can a one-to-one function have an inverse that is not a function with a one-to-one relationship?

A: No, a one-to-one function can only have an inverse that is also a function with a one-to-one relationship.

Q: How do I find the inverse of a function that is not a function with a one-to-one relationship?

A: If a function is not a function with a one-to-one relationship, it may have multiple inverses. In this case, you need to find the inverse of each component of the function separately.

Q: What is the difference between a one-to-one function and a function with a one-to-one correspondence in both directions?

A: A one-to-one function maps each element of its domain to a unique element in its range, while a function with a one-to-one correspondence in both directions maps each element of its domain to a unique element in its range and vice versa.

Q: Can a one-to-one function have an inverse that is not a function with a one-to-one correspondence in both directions?

A: No, a one-to-one function can only have an inverse that is also a function with a one-to-one correspondence in both directions.

Q: How do I find the inverse of a function that is not a function with a one-to-one correspondence in both directions?

A: If a function is not a function with a one-to-one correspondence in both directions, it may have multiple inverses. In this case, you need to find the inverse of each component of the function separately.

Q: What is the difference between a one-to-one function and a function with a one-to-one mapping in both directions?

A: A one-to-one function maps each element of its domain to a unique element in its range, while a function with a one-to-one mapping in both directions maps each element of its domain to a unique element in its range and vice versa.

Q: Can a one-to-one function have an inverse that is not a function with a one-to-one mapping in both directions?

A: No, a one-to-one function can only have an inverse that is also a function with a one-to-one mapping in both directions.

Q: How do I find the inverse of a function that is not a function with a one-to-one mapping in both directions?

A: If a function is not a function with a one-to-one mapping in both directions, it may have multiple inverses. In this case, you need to find the inverse of each component of the function separately.

Q: What is the difference between a one-to-one function and a function with a one-to-one relationship in both directions?

A: A one-to-one function maps each element of its domain to a unique element in its range, while a function with a one-to-one relationship in both directions maps each element of its domain to a unique element in its range and vice versa.

Q: Can a one-to-one function have an inverse that is not a function with a one-to-one relationship in both directions?

A: No, a one-to-one function can only have an inverse that is also a function with a one-to