Suppose [tex]$f(x)$[/tex] is a function such that if [tex]$p\ \textless \ q, f(p)\ \textless \ f(q)$[/tex]. Which statement best describes [tex]$f(x)$[/tex] ?
A. [tex]$f(x)$[/tex] can be odd or even.
B. [tex]$f(x)$[/tex] can be odd but cannot be even.
C. [tex]$f(x)$[/tex] can be even but cannot be odd.
D. [tex]$f(x)$[/tex] cannot be odd or even.
Answer (2)
A strictly increasing function f ( x ) has the property that if p < q , then f ( p ) < f ( q ) .
An even function satisfies f ( − x ) = f ( x ) , and an odd function satisfies f ( − x ) = − f ( x ) .
A strictly increasing function cannot be even because that would imply f ( − x ) = f ( x ) for 0"> x > 0 , contradicting the strictly increasing property.
A strictly increasing function is not necessarily odd. Thus, f ( x ) cannot be odd or even.
f ( x ) cannot be odd or even.
Explanation
Understanding the Problem We are given that f ( x ) is a strictly increasing function, meaning that if p < q , then f ( p ) < f ( q ) . We need to determine whether f ( x ) can be odd, even, both, or neither.
Definitions of Odd and Even Functions Recall the definitions of odd and even functions. A function f ( x ) is even if f ( − x ) = f ( x ) for all x in its domain. A function f ( x ) is odd if f ( − x ) = − f ( x ) for all x in its domain.
Even Functions Let's consider if a strictly increasing function can be even. If f ( x ) is even, then f ( − x ) = f ( x ) . Suppose 0"> x > 0 . Then − x < x . Since f ( x ) is strictly increasing, we must have f ( − x ) < f ( x ) . However, for f ( x ) to be even, we need f ( − x ) = f ( x ) . This is a contradiction. Therefore, a strictly increasing function cannot be even.
Odd Functions Now let's consider if a strictly increasing function can be odd. If f ( x ) is odd, then f ( − x ) = − f ( x ) . Also, if f ( x ) is odd, then f ( 0 ) = − f ( 0 ) , which implies f ( 0 ) = 0 . If 0"> x > 0 , then − x < 0 < x . Since f ( x ) is strictly increasing, we have f ( − x ) < f ( 0 ) < f ( x ) , so f ( − x ) < 0 < f ( x ) . Since f ( x ) is odd, f ( − x ) = − f ( x ) . Thus, − f ( x ) < 0 < f ( x ) , which means 0"> f ( x ) > 0 for 0"> x > 0 . This is possible. For example, f ( x ) = x is strictly increasing and odd. However, this is not the only possibility.
Examples Consider the function f ( x ) = x 3 . This function is strictly increasing and odd. However, consider a function that is strictly increasing but not odd. For example, f ( x ) = e x is strictly increasing, but it is neither odd nor even. Also, consider f ( x ) = x + 100 . This is strictly increasing, but neither odd nor even.
Conclusion Therefore, a strictly increasing function cannot be even, but it is not necessarily odd. Thus, f ( x ) cannot be odd or even.
Examples
In economics, consider a demand function where the quantity demanded increases as the price decreases. This is a strictly decreasing function. Understanding whether such functions can be odd or even helps economists model and analyze market behavior more accurately. For example, knowing that a strictly increasing function cannot be even simplifies the types of models they might consider.
A strictly increasing function cannot be even since it would contradict its increasing property. While it might be odd in some cases, it is not necessarily odd, thus the best choice is that f(x) cannot be classified as odd or even. Therefore, I choose option D: f(x) cannot be odd or even.
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