Let f(x) = x^4 + 2x^2 + 1 / x + 1
Find f(-x)and determine
if f is even, odd, or neither.
f(-x) = x^4 + a x^2 -1/x + 1
so
f(-x) is not the same as f(x) so not even
however f(-x) is not -f(x) either so it is not odd either
Thank you
To find f(-x), we substitute -x into the equation for f(x).
So, we replace x with -x in the function f(x):
f(-x) = (-x)^4 + 2(-x)^2 + 1 / (-x + 1)
Simplifying this expression, we get:
f(-x) = x^4 + 2x^2 + 1 / (-x + 1)
Now, let's compare f(x) and f(-x) to determine if f is even, odd, or neither.
If a function f(x) is even, it satisfies the property f(x) = f(-x) for every x in the domain. In other words, if the input to the function is replaced with its negative, the function output remains the same.
If a function f(x) is odd, it satisfies the property f(x) = -f(-x) for every x in the domain. In other words, if the input to the function is replaced with its negative, the function output changes sign.
If neither of these conditions holds, then the function is neither even nor odd.
Let's check the condition f(x) = f(-x) for f(-x) = x^4 + 2x^2 + 1 / (-x + 1):
f(x) = f(-x) if and only if x^4 + 2x^2 + 1 / (x + 1) = x^4 + 2x^2 + 1 / (-x + 1)
We can see that the denominators (x + 1) and (-x + 1) are different. Therefore, f(x) is neither even nor odd.
Therefore, the function f(x) = x^4 + 2x^2 + 1 / (x + 1) is neither even nor odd.