Given the condition that [tex]\( f(x) \)[/tex] is strictly increasing, meaning that if [tex]\( p < q \)[/tex], then [tex]\( f(p) < f(q) \)[/tex], we need to determine how this property affects whether [tex]\( f(x) \)[/tex] can be classified as an odd or even function.
First, let's recall the definitions of odd and even functions:
- A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex].
- A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex].
Step-by-Step Analysis:
1. Strictly Increasing Function:
- The function [tex]\( f(x) \)[/tex] is strictly increasing, which means [tex]\( f(p) < f(q) \)[/tex] whenever [tex]\( p < q \)[/tex].
- This property tells us that as the input value [tex]\( x \)[/tex] increases, the output value [tex]\( f(x) \)[/tex] also increases.
2. Effect on Odd or Even Nature:
- Even Function: For [tex]\( f(x) \)[/tex] to be even, [tex]\( f(-x) \)[/tex] must equal [tex]\( f(x) \)[/tex]. There is no inherent contradiction here with the function being strictly increasing or decreasing, because this symmetry about the y-axis is a separate aspect from how the function values change as [tex]\( x \)[/tex] increases.
- Odd Function: For [tex]\( f(x) \)[/tex] to be odd, [tex]\( f(-x) \)[/tex] must equal [tex]\( -f(x) \)[/tex]. Similarly, this could happen with a strictly increasing function, as the definition of oddness introduces symmetry about the origin that does not interfere with the monotonic nature of the function.
3. Conclusion:
- The property of being strictly increasing only means that the function's output increases with increasing input. It does not impose any restrictions on whether the function can be classified as odd or even. A strictly increasing function could still have the symmetry properties required to be classified as odd or even, or neither.
Therefore, the best statement that describes [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) \text{ can be odd or even.} \][/tex]
