SOLUTION: Determine algebraically whether the function is even, odd, or neither. g(x)=x^7+8x^5-x^3+6x Need this solved step by step line by line with explanations

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Question 1139191: Determine algebraically whether the function is even, odd, or neither. g(x)=x^7+8x^5-x^3+6x
Need this solved step by step line by line with explanations
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

if you are asked to "determine algebraically" whether a function is even or odd, you take the function and plug -x in for+x, and then simplify
if you end up with the exact+same function that you started with (that is, if f+%28-x%29+=+f+%28x%29, so all of the signs are the same), then the function is even

g%28x%29=x%5E7%2B8x%5E5-x%5E3%2B6x......plug -x in for+x
g%28-x%29=%28-x%29%5E7%2B8%28-x%29%5E5-%28-x%29%5E3%2B6%28-x%29
g%28-x%29=-x%5E7-8x%5E5%2Bx%5E3-6x
=> f+%28-x%29+%3C%3E+f+%28x%29=> a function is not even

for the given function to be odd, I need the above result to have all opposite signs from the original function
So I'll write the original function:
g%28x%29=x%5E7%2B8x%5E5-x%5E3%2B6x....and then switch all the signs
=>-g%28x%29=-x%5E7-8x%5E5%2Bx%5E3-6x=>Comparing this to what I got, I see that they're a match. When I plugged ++-++x in for x, all the signs switched. This means that, as I'd expected:
=> a function is odd


Answer by greenestamps(13215) About Me  (Show Source):
You can put this solution on YOUR website!


(1) Every monomial of odd degree is odd. (x raised to an odd power and (-x) raised to an odd power are opposites.)

(2) Every polynomial that contains only terms of odd degree is odd.

The given function is a polynomial in which the degree of every term is odd; therefore, the function is odd.