Determine whether the function of f(x)=[[x+1]] is odd, even, or neither.
I figured since it's a step function it automatically couldn't have symmetry and therefore is neither.
and doing the whole f(-x) thing:
f(-x)=[[-x+1]]
which is not equal to f(x) or -f(x).
so the answer would be neither, right?
I am not certain of your symbols
If you mean f(x)=absolute(x+1) then it is even, with a line of symettry at x=-1
it's a greatest integer function (AKA step function)
not absolute value.
Yes, you are correct. To determine whether a function is odd, even, or neither, we need to examine its symmetry and determine if it satisfies certain conditions.
Let's consider the function f(x) = [[x + 1]], where [[x]] denotes the greatest integer less than or equal to x. As you mentioned, this is a step function, which means it changes abruptly at certain points.
To determine symmetry, we need to check if the function satisfies the conditions f(-x) = f(x) (even symmetry) or f(-x) = -f(x) (odd symmetry).
For f(x) = [[x + 1]], let's calculate f(-x) and compare it to f(x):
f(-x) = [[-x + 1]]
Since [[x]] is the greatest integer less than or equal to x, in [[-x + 1]], we need to find the greatest integer less than or equal to -x + 1:
If -x + 1 is an integer, then [[-x + 1]] = -x + 1.
If -x + 1 is not an integer, then [[-x + 1]] = -x.
Now let's compare f(-x) to f(x):
If f(-x) = -x + 1, then f(-x) = f(x).
If f(-x) = -x, then f(-x) = -f(x).
Comparing these equations, we can see that f(-x) is not equal to either f(x) or -f(x). Therefore, the function f(x) = [[x + 1]] is neither odd nor even.
Hence, your conclusion that the function is neither odd nor even is correct.