Let's analyze the properties of the function [tex]\( f(x) \)[/tex]. We are given that if [tex]\( p < q \)[/tex], then [tex]\( f(p) < f(q) \)[/tex]. This property tells us that [tex]\( f(x) \)[/tex] is a strictly increasing function, meaning that [tex]\( f(x) \)[/tex] continually increases in value as [tex]\( x \)[/tex] increases.
Now, let’s break down what it means for a function to be even or odd.
1. Even Functions:
A function [tex]\( f(x) \)[/tex] is called even if it satisfies the condition:
[tex]\[
f(x) = f(-x) \quad \text{for all } x.
\][/tex]
2. Odd Functions:
A function [tex]\( f(x) \)[/tex] is called odd if it satisfies the condition:
[tex]\[
f(x) = -f(-x) \quad \text{for all } x.
\][/tex]
Next, let's consider the implications of [tex]\( f(x) \)[/tex] being strictly increasing and either even or odd.
### Case 1: [tex]\( f(x) \)[/tex] is Even
If [tex]\( f(x) \)[/tex] were an even function, then for any [tex]\( x \)[/tex]:
[tex]\[
f(x) = f(-x).
\][/tex]
However, this would imply that [tex]\( f(x) \)[/tex] has the same value at [tex]\( x \)[/tex] and [tex]\( -x \)[/tex], which contradicts the strictly increasing nature of [tex]\( f(x) \)[/tex]. In other words, if [tex]\( x > 0 \)[/tex], then [tex]\( -x < 0 \)[/tex], and strictly increasing would require [tex]\( f(-x) < f(x) \)[/tex], not [tex]\( f(-x) = f(x) \)[/tex]. Therefore, [tex]\( f(x) \)[/tex] cannot be even.
### Case 2: [tex]\( f(x) \)[/tex] is Odd
If [tex]\( f(x) \)[/tex] were an odd function, then for any [tex]\( x \)[/tex]:
[tex]\[
f(x) = -f(-x).
\][/tex]
This condition does not contradict [tex]\( f(x) \)[/tex] being a strictly increasing function. If [tex]\( x > 0 \)[/tex], then [tex]\( -x < 0 \)[/tex], and an increasing function would require [tex]\( f(-x) < f(x) \)[/tex]. In the case of an odd function, [tex]\( f(x) = -f(-x) \)[/tex], if [tex]\( f(-x) < f(x) \)[/tex], then multiplying by [tex]\(-1\)[/tex] maintains consistency with the property of being strictly increasing.
Thus, [tex]\( f(x) \)[/tex] can be odd and still maintain the property of being strictly increasing, but it cannot be even because that would contradict the property of being strictly increasing.
Therefore, the statement that best describes [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) \text{ can be odd but cannot be even.}
\][/tex]
