find an example of an even function f that has the addition property that f (x+15) is odd.
Thanks
To find an example of an even function f that satisfies the given property, let's break it down step by step:
1. Start with an even function: An even function is symmetric with respect to the y-axis, meaning f(-x) = f(x) for all values of x. This means that the function has symmetry across the y-axis.
2. Consider the addition property: The given property states that f(x + 15) is odd. An odd function is symmetric with respect to the origin, meaning f(-x) = -f(x) for all values of x. This means that the function has rotational symmetry of 180 degrees around the origin.
3. Combine the properties: To find an even function f that also satisfies the addition property, we need to find a function that has both symmetry across the y-axis (even function) and symmetry around the origin (odd function).
One example of such a function is f(x) = x.
Explanation:
- This function f(x) = x is an even function because it satisfies f(-x) = f(x) for any value of x.
- To check the addition property, let's evaluate f(x + 15):
f(x + 15) = x + 15
- Now, let's check if this function is odd:
f(-(x + 15)) = - (x + 15) = -x - 15
Since f(x + 15) = -f(-(x + 15)), the function f(x) = x satisfies the addition property required in the question and is also an even function.
Therefore, the function f(x) = x is an example of an even function that has the addition property f(x + 15) = -f(-(x + 15)).