To determine which statement best describes the function [tex]\( f(x) \)[/tex] given that [tex]\( f(p) < f(q) \)[/tex] for [tex]\( p < q \)[/tex], we need to analyze the nature of the function based on the given property.
The condition [tex]\( f(p) < f(q) \)[/tex] for [tex]\( p < q \)[/tex] implies that [tex]\( f(x) \)[/tex] is a strictly increasing function. A strictly increasing function means that as [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] also increases without any decrease.
### Possible Types of Functions
1. Odd Function:
A function [tex]\( f(x) \)[/tex] is considered odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex] in its domain.
2. Even Function:
A function [tex]\( f(x) \)[/tex] is considered even if [tex]\( f(x) = f(-x) \)[/tex] for all [tex]\( x \)[/tex] in its domain.
Let's analyze both possibilities:
Odd Function Analysis:
- Suppose [tex]\( f(x) \)[/tex] is an odd function and strictly increasing.
- For example, [tex]\( f(p) < f(q) \text{ where } p < q \)[/tex], so [tex]\( f(p) \text{ and } f(q) \)[/tex] are positive or negative based on the interval considered.
- Consider [tex]\( f(-x) = -f(x) \)[/tex]:
- If [tex]\( x \)[/tex] increases, then [tex]\(-x\)[/tex] decreases, making [tex]\(-f(x)\)[/tex] decrease as well, which preserves the strictly increasing nature.
- Thus, it's possible for a function to be odd and strictly increasing.
Even Function Analysis:
- Suppose [tex]\( f(x) \)[/tex] is an even function and strictly increasing.
- Since [tex]\( f(x) = f(-x) \)[/tex], the function's value at a positive [tex]\( x \)[/tex] is equal to its value at negative [tex]\( x \)[/tex].
- If the function is strictly increasing for positive [tex]\( x \)[/tex], it must also strictly increase for negative [tex]\( x \)[/tex].
- This would meet the condition [tex]\( f(p) < f(q) \text{ for } p < q \)[/tex] on the negative interval and [tex]\( f(p) < f(q) \text{ for } p < q \)[/tex] on the positive interval.
- Thus, it is also possible for a function to be even and strictly increasing.
### Conclusion
Given that the function [tex]\( f(x) \)[/tex] is strictly increasing, it satisfies the property of increasing without any restrictions on it being odd or even. Hence, we can conclude:
The function [tex]\( f(x) \)[/tex] can be both odd or even.
So, the statement that best describes [tex]\( f(x) \)[/tex] is:
- [tex]\( f(x) \)[/tex] can be odd or even.
Answer:
[tex]\[ \boxed{f(x) \text{ can be odd or even.}} \][/tex]
