SOLUTION: Problem: Determine whether f(x)=3x^4-5x^2+2 is even, odd, or neither. . . I automatically want to say "neither" because of that 3x to the fourth power, but I tried plugging "-x"

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equation Customizable Word Problems -> SOLUTION: Problem: Determine whether f(x)=3x^4-5x^2+2 is even, odd, or neither. . . I automatically want to say "neither" because of that 3x to the fourth power, but I tried plugging "-x"      Log On


   

Question 296771: Problem: Determine whether f(x)=3x^4-5x^2+2 is even, odd, or neither.
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I automatically want to say "neither" because of that 3x to the fourth power, but I tried plugging "-x" in the equation in place of "x" just to test it.
f(-x)=3(-x)^4-5(-x)^2+2.
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My problem is, I don't understand what I'm looking at or why. Should I plug a real number in to test it or what? Any explanation is helpful (especially if I'm doing this wrong!!!) Thanks :)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
A function is even if f%28x%29=f%28-x%29. In other words, plugging in a number will be the same as plugging in the negative of that same number.



f%28x%29=3x%5E4-5x%5E2%2B2 Start with the given function.


f%28-x%29=3%28-x%29%5E4-5%28-x%29%5E2%2B2 Replace each 'x' with '-x'


f%28-x%29=3x%5E4-5%28-x%29%5E2%2B2 Raise -x to the 4th power to get %28-x%29%5E4=%28-x%29%28-x%29%28-x%29%28-x%29=%28-x%29%5E2%28-x%29%5E2=x%5E2%2Ax%5E2=x%5E4


f%28-x%29=3x%5E4-5x%5E2%2B2 Square -x to get x%5E2


So we can see that f%28x%29=f%28-x%29 which means that the function f%28x%29=3x%5E4-5x%5E2%2B2 is an even function. Graphically, this function has symmetry across the y-axis.


Here's a tip: with polynomials, if all of the exponents are even, then the function is even. On the flip side, if all of the exponents are odd, then the function is odd. Note: the constant term has an exponent of 0 (which is even)