Which Function Has An Inverse That Is Also A Function?A. { {(-1,-2),(0,4),(1,3),(5,14),(7,4)}$}$B. { {(-1,2),(0,4),(1,5),(5,4),(7,2)}$}$C. { {(-1,3),(0,4),(1,14),(5,6),(7,2)}$}$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 reverse the operation, i.e., to find the input that corresponds to a given output. However, not all functions have an inverse that is also a function. In this article, we will explore which function among the given options 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 a rule that assigns to each input in the domain a unique output in the range. In other words, a function is a way of mapping each input to a unique output.
What is an Inverse Function?
An inverse function is a function that reverses the operation of the original function. In other words, it is a function that takes the output of the original function and returns the input that produced that output. An inverse function is denoted by the symbol ^{-1}.
When Does a Function Have an Inverse?
A function has an inverse if and only if it is one-to-one, i.e., if each output corresponds to exactly one input. In other words, a function has an inverse if and only if it is injective.
How to Determine if a Function is One-to-One?
To determine if a function is one-to-one, we need to check if each output corresponds to exactly one input. We can do this by checking if the function is either strictly increasing or strictly decreasing.
Strictly Increasing Function
A function is said to be strictly increasing if for any two inputs x and y, if x < y, then f(x) < f(y). In other words, a function is strictly increasing if the output increases as the input increases.
Strictly Decreasing Function
A function is said to be strictly decreasing if for any two inputs x and y, if x < y, then f(x) > f(y). In other words, a function is strictly decreasing if the output decreases as the input increases.
Which Function Has an Inverse That is Also a Function?
Now that we have understood what a function and an inverse function are, let's analyze the given options.
Option A
Option A is given by the set of points {{(-1,-2),(0,4),(1,3),(5,14),(7,4)}$}$. To determine if this function has an inverse that is also a function, we need to check if it is one-to-one.
Let's analyze the points in the set:
- (-1,-2)
- (0,4)
- (1,3)
- (5,14)
- (7,4)
We can see that the output -2 corresponds to two different inputs: -1 and 0. Therefore, this function is not one-to-one, and it does not have an inverse that is also a function.
Option B
Option B is given by the set of points {{(-1,2),(0,4),(1,5),(5,),(7,2)}$}$. To determine if this function has an inverse that is also a function, we need to check if it is one-to-one.
Let's analyze the points in the set:
- (-1,2)
- (0,4)
- (1,5)
- (5,4)
- (7,2)
We can see that the output 2 corresponds to two different inputs: -1 and 7. Therefore, this function is not one-to-one, and it does not have an inverse that is also a function.
Option C
Option C is given by the set of points {{(-1,3),(0,4),(1,14),(5,6),(7,2)}$}$. To determine if this function has an inverse that is also a function, we need to check if it is one-to-one.
Let's analyze the points in the set:
- (-1,3)
- (0,4)
- (1,14)
- (5,6)
- (7,2)
We can see that the output 3 corresponds to only one input: -1. The output 4 corresponds to only one input: 0. The output 14 corresponds to only one input: 1. The output 6 corresponds to only one input: 5. The output 2 corresponds to only one input: 7. Therefore, this function is one-to-one, and it has an inverse that is also a function.
Conclusion
In conclusion, the function that has an inverse that is also a function is Option C, which is given by the set of points {{(-1,3),(0,4),(1,14),(5,6),(7,2)}$}$.
Final Answer
In our previous article, we explored which function among the given options has an inverse that is also a function. We analyzed the options and concluded that Option C, which is given by the set of points {{(-1,3),(0,4),(1,14),(5,6),(7,2)}$}$, has an inverse that is also a function.
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 a rule that assigns to each input in the domain a unique output in the range.
Q: What is an inverse function?
A: An inverse function is a function that reverses the operation of the original function. In other words, it is a function that takes the output of the original function and returns the input that produced that output.
Q: When does a function have an inverse?
A: A function has an inverse if and only if it is one-to-one, i.e., if each output corresponds to exactly one input.
Q: How to determine if a function is one-to-one?
A: To determine if a function is one-to-one, we need to check if each output corresponds to exactly one input. We can do this by checking if the function is either strictly increasing or strictly decreasing.
Q: What is a strictly increasing function?
A: A function is said to be strictly increasing if for any two inputs x and y, if x < y, then f(x) < f(y). In other words, a function is strictly increasing if the output increases as the input increases.
Q: What is a strictly decreasing function?
A: A function is said to be strictly decreasing if for any two inputs x and y, if x < y, then f(x) > f(y). In other words, a function is strictly decreasing if the output decreases as the input increases.
Q: Which function has an inverse that is also a function?
A: The function that has an inverse that is also a function is Option C, which is given by the set of points {{(-1,3),(0,4),(1,14),(5,6),(7,2)}$}$.
Q: Why does Option C have an inverse that is also a function?
A: Option C has an inverse that is also a function because it is one-to-one. Each output in the set corresponds to exactly one input. Therefore, it is possible to reverse the operation of the function, i.e., to find the input that corresponds to a given output.
Q: What are some common examples of functions that have an inverse that is also a function?
A: Some common examples of functions that have an inverse that is also a function include:
- Linear functions: f(x) = ax + b, where a and b are constants.
- Quadratic functions: f(x) = ax^2 + bx + c, where a, b, and c are constants.
- Exponential functions: f(x) = a^x, where a is a constantQ: What are some common examples of functions that do not have an inverse that is also a function?
A: Some common examples of functions that do not have an inverse that is also a function include:
- Constant functions: f(x) = c, where c is a constant.
- Identity functions: f(x) = x.
- Functions with multiple outputs: f(x) = {y1, y2, ...}, where y1, y2, ... are multiple outputs.
Conclusion
In conclusion, we have explored which function among the given options has an inverse that is also a function. We analyzed the options and concluded that Option C, which is given by the set of points {{(-1,3),(0,4),(1,14),(5,6),(7,2)}$}$, has an inverse that is also a function. We also answered some common questions related to functions and inverses.