Is the below function even, odd, or neither

Question

1) f(𝒙) = 𝒙^𝟐+𝟔

2) 𝒈(𝒙) = 𝟕𝒙^𝟑−𝒙

Answer 1

If all terms have x to an even power, the function is even. If all terms have x to an odd power, the function is odd.

Note that a constant is equivalent to x^0 and is even. Also, a linear term, x without a power, is the same as x^1 and is odd.

Thus,

a) even

b) odd

Of course, a plot would tell you that this is the case for both.

The formal way to show even and odd is to show that, in the case of even functions, f(-x) = f(x); in the case of odd functions, f(-x) = -f(x). You clearly get this for a and b.

a) if f(x) = x^2 + 6, f(-x) = (-x)^2 + 6 = x^2 + 6 = f(x); even

b) f(x) = 7x^3 - x; f(-x) = 7(-x)^3 - (-x) = -7x^3 + x = -(7x^3 - x) = -f(x)