Determine whether each of the following functions is even, odd, or neither even nor odd.(a) f(x) = x7 + x (b) g(x) = 1 − x2 (c) h(x) = 2x − x4
Question
Determine whether each of the following functions is even, odd, or neither even nor odd.(a) f(x) = x7 + x (b) g(x) = 1 − x2 (c) h(x) = 2x − x4
Solution
Sure, let's determine whether each function is even, odd, or neither.
(a) f(x) = x^7 + x
A function is even if f(x) = f(-x) and odd if f(-x) = -f(x). Let's check:
f(-x) = (-x)^7 + (-x) = -x^7 - x which is not equal to f(x) = x^7 + x, so it's not even.
Also, -f(x) = -(x^7 + x) = -x^7 - x which is equal to f(-x), so it's odd.
(b) g(x) = 1 - x^2
g(-x) = 1 - (-x)^2 = 1 - x^2 which is equal to g(x), so it's even.
Also, -g(x) = -(1 - x^2) = x^2 - 1 which is not equal to g(-x), so it's not odd.
(c) h(x) = 2x - x^4
h(-x) = 2(-x) - (-x)^4 = -2x - x^4 which is not equal to h(x), so it's not even.
Also, -h(x) = -(2x - x^4) = -2x + x^4 which is not equal to h(-x), so it's not odd.
So, f(x) is odd, g(x) is even, and h(x) is neither even nor odd.
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