Determine if the given function is even, odd or neither.
1. f(x)= 3x^4 -2x^2
Even?
2. f(x)= x^3+x
1 even --> f(x) = f(-x), try x = 3
2 odd --> f(x) = - f(-x) , try x = 3
To determine if a function is even, odd, or neither, we need to consider the properties of even and odd functions.
1. f(x) = 3x^4 - 2x^2
For a function to be even, it must satisfy the condition f(x) = f(-x), where substituting -x for x should give us the same function. Let's check this condition for the given function:
f(-x) = 3(-x)^4 - 2(-x)^2
= 3x^4 - 2x^2
The function f(x) = 3x^4 - 2x^2 is equal to f(-x), satisfying the condition for an even function. Therefore, f(x) is an even function.
2. f(x) = x^3 + x
For a function to be odd, it must satisfy the condition f(x) = -f(-x), where substituting -x for x should give us the opposite function with a negative sign. Let's check this condition for the given function:
f(-x) = (-x)^3 + (-x)
= -x^3 - x
Now let's check if f(x) is equal to the opposite of f(-x):
-f(-x) = -( -x^3 - x)
= x^3 + x
The function f(x) = x^3 + x is equal to the opposite of f(-x), satisfying the condition for an odd function. Therefore, f(x) is an odd function.
In summary:
1. f(x) = 3x^4 - 2x^2 is an even function.
2. f(x) = x^3 + x is an odd function.