Determine if the function is even, odd or neither.
f(x)=x^3-x^2-8x+4
If the function is even,
f(x)=f(-x) for all values of x.
Likewise if the function is odd, f(x)=-f(-x) for all x.
Otherwise, it is neither.
For example,
g(x)=x²
g(-x)=(-x)²=x²
therefore g(x) is even.
h(x)=sin(x)
h(-x)=sin(-x)=-sin(x)
therefore h(x) is odd.
To determine if a function is even, odd, or neither, we need to examine the symmetry of the function.
1. Even function:
A function is considered even if it satisfies the property:
f(x) = f(-x) for all x in the domain.
2. Odd function:
A function is considered odd if it satisfies the property:
f(x) = -f(-x) for all x in the domain.
Now let's apply these properties to the given function, f(x) = x^3 - x^2 - 8x + 4:
1. Checking evenness:
To determine whether the function is even, we need to verify if f(x) = f(-x) for all x in the domain.
Let's substitute -x for x in the function:
f(-x) = (-x)^3 - (-x)^2 - 8(-x) + 4
= -x^3 - x^2 + 8x + 4
We can observe that f(-x) is not equal to f(x) = x^3 - x^2 - 8x + 4. Hence, the function is not even.
2. Checking oddness:
To determine whether the function is odd, we need to verify if f(x) = -f(-x) for all x in the domain.
Let's substitute -x for x in the function:
-f(-x) = -( -x^3 + (-x)^2 + 8(-x) + 4 )
= x^3 - x^2 + 8x - 4
We can observe that f(x) is not equal to -f(-x). Therefore, the given function is not odd.
Since the function does not satisfy the criteria for evenness or oddness, we conclude that the given function f(x) = x^3 - x^2 - 8x + 4 is neither even nor odd.