Module 2: Algebraically Determining Even/Odd Function (Rational Example)

[@internallyinteral]

To determine whether the given function f(x)=−2x3−x3x4+5x2f(x)=−3x4+5x22x3−x​ is even, odd, or neither, we can use the definitions of these types of functions: An even function is one for which f(x)=f(−x)f(x)=f(−x) for all xx in its domain. An odd function is one for which f(x)=−f(−x)f(x)=−f(−x) for all xx in its domain. Let's check each condition: Even Function Test: f(x)=−2x3−x3x4+5x2f(x)=−3x4+5x22x3−x​ f(−x)=−2(−x)3−(−x)3(−x)4+5(−x)2f(−x)=−3(−x)4+5(−x)22(−x)3−(−x)​ Simplifying both numerator and denominator: f(−x)=−−2x3+x3x4+5x2f(−x)=−3x4+5x2−2x3+x​ We can see that f(x)f(x) is NOT equal to f(−x)f(−x) because the signs are different in the numerators. Therefore, f(x)f(x) is NOT an even function. Odd Function Test: f(x)=−2x3−x3x4+5x2f(x)=−3x4+5x22x3−x​ f(−x)=−2(−x)3−(−x)3(−x)4+5(−x)2f(−x)=−3(−x)4+5(−x)22(−x)3−(−x)​ Simplifying both numerator and denominator: f(−x)=−−2x3+x3x4+5x2f(−x)=−3x4+5x2−2x3+x​ We can see that f(−x)f(−x) is equal to −f(x)−f(x) because they have the same value but with a negative sign. Therefore, f(x)f(x) is an odd function. In conclusion, the given function f(x)=−2x3−x3x4+5x2f(x)=−3x4+5x22x3−x​ is an odd function.

[@dapp_er]

The answer is wrong plus the steps you didn't change the signs of the whole function when you factored it by -1;The answer should be Neither odd nor even...(WRONG ANSWER)!!!

[@megantruong2062]

It’s right

[@christinat4507]

that's what I though too.

[@Etk333]

Wouldnt it be odd. The numerator negative cancels with the negative equatione.Thus you factor a negative and numerator is back to 2x^3-x