Which Function Has An Inverse That Is Also A Function?A. \[$\{(-4,3),(-2,7),(-1,0),(4,-3),(11,-7)\}\$\]B. \[$\{(-4,6),(-2,2),(-1,6),(4,2),(11,2)\}\$\]C. \[$\{(-4,5),(-2,9),(-1,8),(4,8),(11,4)\}\$\]D.
In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). A function is said to have an inverse if it is possible to "undo" the function, i.e., to find the input that corresponds to a given output. However, not all functions have inverses that are also functions. In this article, we will explore which of the given functions has an inverse that is also a function.
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). It is often represented as a set of ordered pairs, where each pair consists of an input and an output. For example, the function f(x) = 2x can be represented as the set of ordered pairs {(1, 2), (2, 4), (3, 6), ...}.
What is an Inverse Function?
An inverse function is a function that "undoes" another function. In other words, if we have a function f(x) and an inverse function f^(-1)(x), then f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. This means that if we apply the inverse function to the output of the original function, we get back the original input.
Which Functions Have Inverses That Are Also Functions?
Not all functions have inverses that are also functions. In fact, a function must satisfy certain conditions in order to have an inverse that is also a function. These conditions are:
- One-to-one: The function must be one-to-one, meaning that each output corresponds to exactly one input.
- Onto: The function must be onto, meaning that every possible output is reached.
- Injective: The function must be injective, meaning that each output corresponds to exactly one input.
Analyzing the Options
Now that we have a good understanding of what a function and an inverse function are, let's analyze the options given in the problem.
Option A
Option A is the set of ordered pairs {(-4,3),(-2,7),(-1,0),(4,-3),(11,-7)}. To determine if this function has an inverse that is also a function, we need to check if it satisfies the conditions of being one-to-one, onto, and injective.
- One-to-one: Each output corresponds to exactly one input. For example, the output 3 corresponds to the input -4, and the output 7 corresponds to the input -2.
- Onto: Every possible output is reached. In this case, the outputs are 3, 7, 0, -3, and -7, which are all reached.
- Injective: Each output corresponds to exactly one input. This is also true, as each output corresponds to exactly one input.
Therefore, option A satisfies all the conditions and has an inverse that is also a function.
Option B
Option B is the set of ordered pairs {(-4,6),(-2,2),(-1,6),(4,2),(11,2)}. To determine if this function has an inverse that is also a function, we need to check if it satisfies the conditions of being one-to-one, onto, and injective.
- One-to-one: Each output corresponds to exactly one input. However, the output 6 corresponds to two inputs, -4 and -1.
- Onto: Every possible output is reached. In this case, the outputs are 6, 2, which are not all reached.
- Injective: Each output corresponds to exactly one input. This is not true, as the output 6 corresponds to two inputs.
Therefore, option B does not satisfy all the conditions and does not have an inverse that is also a function.
Option C
Option C is the set of ordered pairs {(-4,5),(-2,9),(-1,8),(4,8),(11,4)}. To determine if this function has an inverse that is also a function, we need to check if it satisfies the conditions of being one-to-one, onto, and injective.
- One-to-one: Each output corresponds to exactly one input. However, the output 8 corresponds to two inputs, -1 and 4.
- Onto: Every possible output is reached. In this case, the outputs are 5, 9, 8, 4, which are not all reached.
- Injective: Each output corresponds to exactly one input. This is not true, as the output 8 corresponds to two inputs.
Therefore, option C does not satisfy all the conditions and does not have an inverse that is also a function.
Conclusion
In conclusion, only option A satisfies all the conditions of being one-to-one, onto, and injective, and therefore has an inverse that is also a function.
Final Answer
In the previous article, we discussed which function has an inverse that is also a function. We analyzed the options given and determined that only option A satisfies all the conditions of being one-to-one, onto, and injective, and therefore has an inverse that is also a function. In this article, we will answer some frequently asked questions about functions and inverses.
Q: What is a function?
A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). It is often represented as a set of ordered pairs, where each pair consists of an input and an output.
Q: What is an inverse function?
A: An inverse function is a function that "undoes" another function. In other words, if we have a function f(x) and an inverse function f^(-1)(x), then f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. This means that if we apply the inverse function to the output of the original function, we get back the original input.
Q: What are the conditions for a function to have an inverse that is also a function?
A: A function must satisfy the following conditions to have an inverse that is also a function:
- One-to-one: The function must be one-to-one, meaning that each output corresponds to exactly one input.
- Onto: The function must be onto, meaning that every possible output is reached.
- Injective: The function must be injective, meaning that each output corresponds to exactly one input.
Q: How do I determine if a function has an inverse that is also a function?
A: To determine if a function has an inverse that is also a function, you need to check if it satisfies the conditions of being one-to-one, onto, and injective. You can do this by analyzing the ordered pairs that make up the function.
Q: What is the difference between a function and an inverse function?
A: A function is a relation between a set of inputs and a set of possible outputs, while an inverse function is a function that "undoes" another function. In other words, a function takes an input and produces an output, while an inverse function takes an output and produces an input.
Q: Can a function have multiple inverses?
A: No, a function cannot have multiple inverses. An inverse function is unique and is the only function that "undoes" the original function.
Q: Can a function have an inverse that is not a function?
A: Yes, a function can have an inverse that is not a function. This occurs when the function is not one-to-one, onto, or injective.
Q: What is the significance of having an inverse that is also a function?
A: Having an inverse that is also a function is significant because it means that the function is bijective, meaning that it is both one-to-one and onto. This is a desirable property in many mathematical and real-world applications.
Q: Can you give an example of a function that has an inverse that is also a function?
A: Yes, the function f(x) = 2x has an inverse that is also a function, which is f^(-1)(x) = x/2.
Q: Can you give an example of a function that does not have an inverse that is also a function?
A: Yes, the function f(x) = x^2 has an inverse that is not a function, which is f^(-1)(x) = ±√x.
Conclusion
In conclusion, having an inverse that is also a function is a desirable property in many mathematical and real-world applications. We hope that this Q&A article has helped to clarify the concepts of functions and inverses, and has provided a better understanding of the conditions for a function to have an inverse that is also a function.