Answer :
To determine why the function [tex]\( f(x) = 2x - 3 \)[/tex] has an inverse relation that is also a function, let's analyze the given statements:
1. The graph of [tex]\( f(x) \)[/tex] passes the vertical line test.
2. [tex]\( f(x) \)[/tex] is a one-to-one function.
3. The graph of the inverse of [tex]\( f(x) \)[/tex] passes the horizontal line test.
4. [tex]\( f(x) \)[/tex] is not a function.
### Explanation:
For a function to have an inverse that is also a function, it must be both a function and one-to-one.
1. Vertical Line Test: This tests if a graph is a function. Each vertical line should intersect the graph of the function at most once. The function [tex]\( f(x) = 2x - 3 \)[/tex] passes the vertical line test, confirming that [tex]\( f(x) \)[/tex] is indeed a function. However, this does not guarantee that its inverse is also a function.
2. One-to-One Function: A function is one-to-one if and only if every horizontal line intersects the graph of the function at most once. This can also be checked by ensuring that the function is strictly increasing or decreasing. The function [tex]\( f(x) = 2x - 3 \)[/tex] is a linear function with a positive slope (2), meaning it is strictly increasing. Therefore, it is one-to-one.
3. Horizontal Line Test for the Inverse: This tests if the inverse relation is a function. If every horizontal line only intersects the graph of [tex]\( f(x) \)[/tex] at most once, then the inverse of [tex]\( f(x) \)[/tex] will pass the vertical line test, making the inverse a function as well. Since [tex]\( f(x) = 2x - 3 \)[/tex] is strictly increasing, it passes the horizontal line test, meaning its inverse will indeed be a function.
4. [tex]\( f(x) \)[/tex] is not a Function: This statement is false as [tex]\( f(x) = 2x - 3 \)[/tex] is clearly a function.
### Conclusion:
For [tex]\( f(x) = 2x - 3 \)[/tex] to have an inverse that is a function, it must be a one-to-one function. Therefore, the correct statement is:
[tex]\[ \boxed{f(x) \text{ is a one-to-one function.}} \][/tex]
That corresponds to statement [tex]\( \boxed{2} \)[/tex].
1. The graph of [tex]\( f(x) \)[/tex] passes the vertical line test.
2. [tex]\( f(x) \)[/tex] is a one-to-one function.
3. The graph of the inverse of [tex]\( f(x) \)[/tex] passes the horizontal line test.
4. [tex]\( f(x) \)[/tex] is not a function.
### Explanation:
For a function to have an inverse that is also a function, it must be both a function and one-to-one.
1. Vertical Line Test: This tests if a graph is a function. Each vertical line should intersect the graph of the function at most once. The function [tex]\( f(x) = 2x - 3 \)[/tex] passes the vertical line test, confirming that [tex]\( f(x) \)[/tex] is indeed a function. However, this does not guarantee that its inverse is also a function.
2. One-to-One Function: A function is one-to-one if and only if every horizontal line intersects the graph of the function at most once. This can also be checked by ensuring that the function is strictly increasing or decreasing. The function [tex]\( f(x) = 2x - 3 \)[/tex] is a linear function with a positive slope (2), meaning it is strictly increasing. Therefore, it is one-to-one.
3. Horizontal Line Test for the Inverse: This tests if the inverse relation is a function. If every horizontal line only intersects the graph of [tex]\( f(x) \)[/tex] at most once, then the inverse of [tex]\( f(x) \)[/tex] will pass the vertical line test, making the inverse a function as well. Since [tex]\( f(x) = 2x - 3 \)[/tex] is strictly increasing, it passes the horizontal line test, meaning its inverse will indeed be a function.
4. [tex]\( f(x) \)[/tex] is not a Function: This statement is false as [tex]\( f(x) = 2x - 3 \)[/tex] is clearly a function.
### Conclusion:
For [tex]\( f(x) = 2x - 3 \)[/tex] to have an inverse that is a function, it must be a one-to-one function. Therefore, the correct statement is:
[tex]\[ \boxed{f(x) \text{ is a one-to-one function.}} \][/tex]
That corresponds to statement [tex]\( \boxed{2} \)[/tex].
