To determine which statement best describes the function [tex]\( f(x) \)[/tex] under the condition that if [tex]\( p < q \)[/tex], then [tex]\( f(p) < f(q) \)[/tex], let's analyze the properties of the function step-by-step.
1. Understanding the Given Condition:
- The condition [tex]\( p < q \implies f(p) < f(q) \)[/tex] means that the function [tex]\( f(x) \)[/tex] is strictly increasing. This means that for any two points [tex]\( p \)[/tex] and [tex]\( q \)[/tex] where [tex]\( p < q \)[/tex], the value of [tex]\( f(p) \)[/tex] will always be less than [tex]\( f(q) \)[/tex].
2. Evaluating Even Functions:
- A function [tex]\( f(x) \)[/tex] is considered even if [tex]\( f(x) = f(-x) \)[/tex] for all [tex]\( x \)[/tex]. This implies symmetry about the y-axis.
- Suppose [tex]\( f(x) \)[/tex] were even. Then for any [tex]\( x \)[/tex], [tex]\( f(x) \)[/tex] would equal [tex]\( f(-x) \)[/tex]. However, since [tex]\( f(x) \)[/tex] is strictly increasing, if [tex]\( x > 0 \)[/tex], then [tex]\( f(x) > f(0) \)[/tex] and [tex]\( f(-x) < f(0) \)[/tex]. This contradicts the property of even functions. Therefore, a strictly increasing function cannot be even.
3. Evaluating Odd Functions:
- A function [tex]\( f(x) \)[/tex] is considered odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex]. This implies symmetry about the origin.
- Suppose [tex]\( f(x) \)[/tex] were odd. Then for any [tex]\( x \)[/tex], [tex]\( f(-x) \)[/tex] would equal [tex]\( -f(x) \)[/tex]. However, since [tex]\( f(x) \)[/tex] is strictly increasing, if [tex]\( x > 0 \)[/tex], then [tex]\( f(x) > f(0) \)[/tex] and [tex]\( -f(x) < f(0) \)[/tex]. This also contradicts the strictly increasing property because [tex]\( f(-x) \)[/tex] should be strictly less than [tex]\( f(0) \)[/tex] for [tex]\( x > 0 \)[/tex]. Therefore, a strictly increasing function cannot be odd.
4. Conclusion:
- Since neither the even nor the odd condition can be satisfied by a strictly increasing function [tex]\( f(x) \)[/tex], we conclude that the function [tex]\( f(x) \)[/tex] cannot be odd or even.
Therefore, the best statement that describes [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) \text{ cannot be odd or even.} \][/tex]
Thus, the correct statement is:
[tex]\[ \boxed{f(x) \text{ cannot be odd or even.}} \][/tex]
