How do you determine if #f(x)=4x^3# is an even or odd function?

Question

Anna Allen · Accepted Answer

#f(x) = 4x^3# is an odd function.

An even function is one for which #f(-x) = f(x)# for all #x# in its domain. An odd function is one for which #f(-x) = -f(x)# for all #x# in its domain. In our illustration: #f(-x) = 4(-x)^3 = -4x^3 = -f(x)# for all values of #x#. So #f(x) = 4x^3# is an odd function. #color(white)()# Footnote There is a quick way to determine if a polynomial is odd or even: Do any terms have an odd degree, an even degree, or a combination? If the function is odd, then it is odd; if it is even, then it is even; and if it is neither, then it is neither. Note that constant terms are of even (#0#) degree.

Serenity Jones · Answer

odd function

Examine the following to see if a function f(x) is even or odd. • f(x) is even if f(x) = f( -x) About the y-axis, even functions are symmetrical. • f(x) is odd if - f(x) = f(-x) There is symmetry around the origin of odd functions. Check for even function. #f(-x)=4(-x)^3=-4x^3≠f(x)# f(x) is not even since f(x) ≠ f(-x) Check for strange functions #-f(x)=-(4x^3)=-4x^3=f(-x)# f(x) is odd since - f(x) = f( -x) graph{4x^3 [-10, 10, -5, 5]} Take note of the origin's symmetry.

Cora Alexander · Answer

undefined To determine if a function $ f(x) = 4x^3 $ is even or odd:

1. Even function: $ f(x) $ is even if $ f(-x) = f(x) $ for all $ x $.
2. Odd function: $ f(x) $ is odd if $ f(-x) = -f(x) $ for all $ x $.

Let's evaluate $ f(-x) $ for $ f(x) = 4x^3 $:

$$ f(-x) = 4(-x)^3 = -4x^3 $$

Comparing $ f(-x) $ to $ f(x) = 4x^3 $:

1. If $ f(-x) = f(x) $, the function is even. But $ -4x^3 $ is not equal to $ 4x^3 $, so $ f(x) = 4x^3 $ is not even.
2. If $ f(-x) = -f(x) $, the function is odd. And since $ -4x^3 = -4x^3 $, the function $ f(x) = 4x^3 $ is odd.