f(x) = {(1, 4), (2, 3), (5, 8), (4, 7)} Is f-1(x) a function and why?

Yes, each element in the range has only one domain value.
No, we don’t know the original function and therefore can’t make the determination.
Yes, each element in the domain has only one range value.
No, there are only a limited number of ordered pairs and can’t be determined.

The correct answer is:

**Yes, each element in the range has only one domain value.**

### Explanation:
For \( f(x) \) to have an inverse that is also a function, each element in the range (output values) must correspond to exactly one element in the domain (input values). 

Given the function \( f(x) = \{(1, 4), (2, 3), (5, 8), (4, 7)\} \):
- The range (outputs) is \( \{4, 3, 8, 7\} \).
- The domain (inputs) is \( \{1, 2, 5, 4\} \).

When checking if \( f^{-1}(x) \) is a function, we look at the elements in the range of \( f(x) \) to ensure each corresponds to exactly one element in the domain. Since this is true for all the pairs, \( f^{-1}(x) \) is indeed a function.

### Relative Questions:
1. What is the general criterion for a function to have an inverse that is also a function?
2. How do you find the inverse of a function given a set of ordered pairs?
3. If the function \( f(x) \) were given by \( f(x) = \{(1, 4), (2, 4), (5, 8), (4, 7)\} \), would \( f^{-1}(x) \) be a function? Why or why not?
4. What would the inverse \( f^{-1}(x) \) look like for the given set of ordered pairs?
5. Can a function have more than one inverse? Why or why not?

### Tip:
When determining if the inverse of a function is also a function, always check that each output value in the original function corresponds to exactly one input value.

f(x) = {(1, 4), (2, 3), (5, 8), (4, 7)} Is f-1(x) a function and why?

Explore whether the inverse of the function f(x) = {(1, 4), (2, 3), (5, 8), (4, 7)} is a function. This problem involves concepts related to functions and their inverses, suitable for students in Grades 9-12.

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