Finding the Function Inverse

Oct 13, 2025 by miodeya.com 51 views

The Function Inverse: Understanding h(x) and h^(-1)(x)

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. The function is often denoted by a symbol, such as f(x) or h(x), where x represents the input and the function returns a corresponding output. In this article, we will explore the concept of a function inverse, denoted by h^(-1)(x), and how it relates to the original function h(x).

What is a Function?

A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. The function is often denoted by a symbol, such as f(x) or h(x), where x represents the input and the function returns a corresponding output. For example, the function h(x) = 2x + 3 takes an input x and returns an output that is twice the input plus 3.

The Function h(x)

The function h(x) is given by the following set of ordered pairs:

h(x)=\{(3,-5),(5,-7),(6,-9),(10,-12),(12,-16)\}

This means that when the input x is 3, the output is -5, when the input x is 5, the output is -7, and so on.

What is a Function Inverse?

A function inverse, denoted by h^(-1)(x), is a relation that undoes the action of the original function h(x). In other words, if h(x) takes an input x and returns an output y, then h^(-1)(x) takes the output y and returns the original input x.

Finding the Function Inverse

To find the function inverse h^(-1)(x), we need to swap the x and y values in the original function h(x). This means that we need to take the output y and make it the input x, and vice versa.

Option A: h^(-1)(x) = {(3,5),(5,7),(6,9),(10,12),(12,16)}

Option A gives the following set of ordered pairs:

\{(3,5),(5,7),(6,9),(10,12),(12,16)\}

This means that when the input x is 3, the output is 5, when the input x is 5, the output is 7, and so on.

Option B: h^(-1)(x) = {(5,3),(7,5),(9,6),(12,10),(16,12)}

Option B gives the following set of ordered pairs:

\{(5,3),(7,5),(9,6),(12,10),(16,12)\}

This means that when the input x is 5, the output is 3, when the input x is 7, the output is 5, and so on.

In conclusion, the function inverse h^(-1)(x) is a relation that undoes the action of the original function h(x). To find the function inverse, we need to swap the x and y values in the original function h(x). In this article, we explored the concept of a function inverse and how it relates to the original function h(x). We also examined two possible options for the function inverse h^(-1)(x) and determined which one is correct.

The correct answer is Option A: h^(-1)(x) = {(3,5),(5,7),(6,9),(10,12),(12,16)}.

Why is Option A Correct?

Option A is correct because it correctly swaps the x and y values in the original function h(x). This means that when the input x is 3, the output is 5, when the input x is 5, the output is 7, and so on. This is the definition of a function inverse, and it is the only option that satisfies this definition.

Why is Option B Incorrect?

Option B is incorrect because it does not correctly swap the x and y values in the original function h(x). This means that when the input x is 5, the output is 3, when the input x is 7, the output is 5, and so on. This is not the definition of a function inverse, and it is not the correct answer.

In our previous article, we explored the concept of a function inverse, denoted by h^(-1)(x), and how it relates to the original function h(x). We also examined two possible options for the function inverse h^(-1)(x) and determined which one is correct. In this article, we will answer some frequently asked questions about function inverses and provide additional examples to help solidify your understanding.

Q: What is the purpose of a function inverse?

A: The purpose of a function inverse is to undo the action of the original function. In other words, if h(x) takes an input x and returns an output y, then h^(-1)(x) takes the output y and returns the original input x.

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

A: To find the function inverse of a given function, you need to swap the x and y values in the original function. This means that you need to take the output y and make it the input x, and vice versa.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two related but distinct concepts. A function takes an input x and returns an output y, while its inverse takes the output y and returns the original input x.

Q: Can a function have more than one inverse?

A: No, a function can only have one inverse. The inverse of a function is a unique relation that undoes the action of the original function.

Q: Can a function have no inverse?

A: Yes, a function can have no inverse. This occurs when the function is not one-to-one, meaning that it maps multiple inputs to the same output.

Q: What is an example of a function that has no inverse?

A: An example of a function that has no inverse is the function f(x) = x^2. This function maps multiple inputs to the same output, making it impossible to find a unique inverse.

Q: Can a function have multiple inverses?

A: No, a function can only have one inverse. However, a function can have multiple functions that are inverses of each other.

Q: What is an example of a function that has multiple inverses?

A: An example of a function that has multiple inverses is the function f(x) = x^2. While this function has no unique inverse, it has multiple functions that are inverses of each other, such as f^(-1)(x) = ±√x.

In conclusion, the function inverse h^(-1)(x) is a relation that undoes the action of the original function h(x). To find the function inverse, you need to swap the x and y values in the original function. In this article, we answered some frequently asked questions about function inverses and provided additional examples to help solidify your understanding.

Here are some additional examples of functions and their inverses:

f(x) = 2x + 3: h^(-1)(x) = (x - 3)/2
f(x) = x^2: h^(-1)(x) = ±√x
f(x) = 3x - 2: h^(-1)(x) = (x + 2)/3

In conclusion, the function inverse h^(-1)(x) is a powerful tool for understanding and working with functions. By swapping the x and y values in the original function, you can find the inverse of a function and use it to solve problems and simplify expressions. We hope this article has helped you understand the concept of function inverses and how to apply them in practice.