Question
Must a function that is decreasing over a given interval always be negative over that same interval? Explain.
Answers
GPT-4o mini
No, a function that is decreasing over a given interval does not have to be negative over that same interval.
A decreasing function is defined as a function \( f(x) \) such that for any two points \( a \) and \( b \) in the interval, if \( a < b \), then \( f(a) \geq f(b) \). This means that the function values are getting smaller as you move from left to right across the interval.
However, this does not impose any restrictions on the actual values of the function. A decreasing function can have positive, negative, or zero values over the interval. For instance:
1. **Example of a Positive Decreasing Function**: Consider the function \( f(x) = 5 - x \) over the interval \( [0, 5] \). The function decreases from \( f(0) = 5 \) to \( f(5) = 0 \), and during the interval, it remains positive for \( x \in [0, 5) \) but reaches zero at \( x = 5 \).
2. **Example of a Negative Decreasing Function**: An example of a decreasing function that is entirely negative could be \( f(x) = -x - 1 \) over the interval \( [0, 2] \). This function decreases from \( f(0) = -1 \) to \( f(2) = -3 \).
3. **Example of a Mixed Behavior**: Consider \( f(x) = -x^2 + 4 \) over the interval \( [-3, 2] \). This function is positive for some parts of the interval, decreases from \( 4 \) at \( x = 0 \) down to \( 0 \) at \( x = 2 \), and will also take on negative values for \( x > 2 \).
In summary, while a decreasing function may take on negative values at some points, it is not a requirement for the function to be negative over the entire interval where it is decreasing. Decreasing behavior is about the relationship between \( f(a) \) and \( f(b) \) rather than the sign of the function values themselves.
A decreasing function is defined as a function \( f(x) \) such that for any two points \( a \) and \( b \) in the interval, if \( a < b \), then \( f(a) \geq f(b) \). This means that the function values are getting smaller as you move from left to right across the interval.
However, this does not impose any restrictions on the actual values of the function. A decreasing function can have positive, negative, or zero values over the interval. For instance:
1. **Example of a Positive Decreasing Function**: Consider the function \( f(x) = 5 - x \) over the interval \( [0, 5] \). The function decreases from \( f(0) = 5 \) to \( f(5) = 0 \), and during the interval, it remains positive for \( x \in [0, 5) \) but reaches zero at \( x = 5 \).
2. **Example of a Negative Decreasing Function**: An example of a decreasing function that is entirely negative could be \( f(x) = -x - 1 \) over the interval \( [0, 2] \). This function decreases from \( f(0) = -1 \) to \( f(2) = -3 \).
3. **Example of a Mixed Behavior**: Consider \( f(x) = -x^2 + 4 \) over the interval \( [-3, 2] \). This function is positive for some parts of the interval, decreases from \( 4 \) at \( x = 0 \) down to \( 0 \) at \( x = 2 \), and will also take on negative values for \( x > 2 \).
In summary, while a decreasing function may take on negative values at some points, it is not a requirement for the function to be negative over the entire interval where it is decreasing. Decreasing behavior is about the relationship between \( f(a) \) and \( f(b) \) rather than the sign of the function values themselves.