Let's analyze the problem step-by-step:
1. Understanding the Function Property:
We are given that for the function [tex]\( f(x) \)[/tex], if [tex]\( p < q \)[/tex], then [tex]\( f(p) < f(q) \)[/tex]. This implies that [tex]\( f(x) \)[/tex] is a strictly increasing function. In other words, as [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] also increases, and there are no two distinct values [tex]\( p \)[/tex] and [tex]\( q \)[/tex] where [tex]\( f(p) = f(q) \)[/tex] if [tex]\( p \neq q \)[/tex].
2. Criteria of Even and Odd Functions:
- Even Function: A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex]. This means the graph of [tex]\( f(x) \)[/tex] is symmetric about the y-axis.
- Odd Function: A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex]. This means the graph of [tex]\( f(x) \)[/tex] is symmetric about the origin.
3. Evaluation of Conditions:
- Strictly Increasing and Even: If [tex]\( f(x) \)[/tex] is strictly increasing and even, then for positive values as [tex]\( x \)[/tex] increases [tex]\( f(x) \)[/tex] must increase. For the negative values, since [tex]\( f(-x) = f(x) \)[/tex], [tex]\( f(x) \)[/tex] values must mirror the positive side which will maintain strict increase on both sides.
- Strictly Increasing and Odd: If [tex]\( f(x) \)[/tex] is strictly increasing and odd, then [tex]\( f(-x) = -f(x) \)[/tex]. This also fits within the constraints, as it merely reverses the axis but does not conflict with the property of being strictly increasing.
4. Conclusion:
Given the interpretations and properties discussed above, we can infer that [tex]\( f(x) \)[/tex] can be either an even function or an odd function, fulfilling the requirement of the problem being strictly increasing in both conditions. Hence:
[tex]\[ \text{The correct statement is: } f(x) \text{ can be odd or even.} \][/tex]
Thus, the best description for [tex]\( f(x) \)[/tex] given its strictly increasing property is:
[tex]\[ \boxed{f(x) \text{ can be odd or even.}} \][/tex]
