Answer: No, it is not possible.
Explanation:
No, it is not possible for a function to be both even and odd.
To understand why, let's first define what it means for a function to be even or odd.
- An even function is symmetric with respect to the y-axis. This means that if you reflect any point on the graph of an even function across the y-axis, you will get another point on the graph of the function. In other words, if we have a function f(x), then f(x) = f(-x) for all x in the domain of the function.
- On the other hand, an odd function is symmetric with respect to the origin. This means that if you reflect any point on the graph of an odd function across the origin, you will get another point on the graph of the function. In other words, if we have a function f(x), then f(x) = -f(-x) for all x in the domain of the function.
Now, let's consider a function that is both even and odd. This would mean that for every value of x, f(x) = f(-x) and f(x) = -f(-x).
If we set x = -x, we get f(-x) = f(x).
Now, if we substitute this back into the definition of an odd function, we get f(x) = -f(-x) = -f(x).
So, we have f(x) = -f(x), which means that for every value of x, f(x) is equal to its negative value.
The only function that satisfies this condition is the zero function, where f(x) = 0 for all values of x.
Therefore, the only function that can be both even and odd is the zero function.
In summary, a function cannot be both even and odd, except for the zero function.
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