Question

1) g(x)=2x^4-x^2; 2) f(x)=x^3+2 Четная или нечетная функция? У меня получилось так: 1) g(-x)=2(-x)^4-(-x)^2=2x^4-x=g(x) она четная; 2) f(-x)=(-x)^3+2=-x^3+2=-f(x) нечетная

Answer 1

1) g(x)=2x^4-x^2; 2) f(x)=x^3+2

Четная или нечетная функция?

1) g(-x)=2(-x)^4-(-x)^2=2x^4-x^2=g(x) она четная;

2) f(-x)=(-x)^3+2=-x^3+2 (не равна -f(x) !!! - минус только перед первым членом)

эта функция ни четная ни нечетная!!!

Answer 2

Your analysis is almost correct! Let's go through it step by step:

1. For the function g(x) = 2x^4 - x^2: To check for evenness, we need to verify whether g(-x) = g(x).

g(-x) = 2(-x)^4 - (-x)^2 = 2x^4 - x^2 = g(x)

Since g(-x) = g(x), the function g(x) is even.

2. For the function f(x) = x^3 + 2: To check for oddness, we need to verify whether f(-x) = -f(x).

f(-x) = (-x)^3 + 2 = -x^3 + 2

However, you made a slight mistake here: -f(x) = -(x^3 + 2) = -x^3 - 2

Now, compare f(-x) and -f(x):

f(-x) = -x^3 + 2 -f(x) = -x^3 - 2

As we can see, f(-x) is not equal to -f(x), which means the function f(x) is not odd.

In conclusion:

1. The function g(x) is even.
2. The function f(x) is neither even nor odd.