Question
No, a function that is decreasing over a given interval does not have to be negative over that interval.
A decreasing function means that as the input (or x-value) increases, the output (or y-value) decreases. However, the values of the function can be positive, negative, or zero at different points within that interval.
A decreasing function means that as the input (or x-value) increases, the output (or y-value) decreases. However, the values of the function can be positive, negative, or zero at different points within that interval.
Answers
GPT-4o mini
You are correct. A decreasing function is characterized by the property that as the input (x) increases, the output (f(x)) decreases. However, this decrease in output does not imply that the function must take on negative values within a specified interval.
For example, consider the function \( f(x) = 1 - x \). This function is decreasing over all real numbers. For values of \( x \) less than 1 (e.g., \( x = 0 \), where \( f(0) = 1 \)), \( f(x) \) is positive. However, as \( x \) approaches 1, \( f(x) \) decreases to 0, and for \( x > 1 \), \( f(x) \) becomes negative.
So, a decreasing function can have a mix of positive, negative, and zero values over its interval. The important takeaway is that the defining feature of a decreasing function pertains to the relationship between the inputs and corresponding outputs, rather than the sign of the outputs themselves.
For example, consider the function \( f(x) = 1 - x \). This function is decreasing over all real numbers. For values of \( x \) less than 1 (e.g., \( x = 0 \), where \( f(0) = 1 \)), \( f(x) \) is positive. However, as \( x \) approaches 1, \( f(x) \) decreases to 0, and for \( x > 1 \), \( f(x) \) becomes negative.
So, a decreasing function can have a mix of positive, negative, and zero values over its interval. The important takeaway is that the defining feature of a decreasing function pertains to the relationship between the inputs and corresponding outputs, rather than the sign of the outputs themselves.