I don't understand how you determine whether a function is even, odd,or neither.

Here are my problems:

Determine whether the given function is even, odd, or neither.

8. f(x)=x^3-x^2
This is how I did it.
f(-x)=(-x^3)- (-x^2)= (-x)(-x)(-x) - (-x)(-x) = (-x^3) -(x^2)- I got that it was odd

9. f(x)=-4x^5+x^3 This is how I did it:
f(-x)=-4(-x^5) + (-x^3)=4x^5-x^3 - I got that it was neither.

8. Both answers are wrong. The #8 f(x) is neither even nor odd. It is the sum of an even term (x^2) and and odd term (x^3). The #9 f(x) is odd. It is the sum of two odd terms.

To determine whether a function is even, odd, or neither, you need to evaluate the function using the property of even and odd functions.

1. Even Function: A function f(x) is even if f(x) = f(-x) for all values of x.

2. Odd Function: A function f(x) is odd if f(x) = -f(-x) for all values of x.

3. Neither: If the function does not satisfy either of the above conditions, it is neither even nor odd.

Now, let's apply these concepts to your given functions:

8. f(x) = x^3 - x^2

To determine whether this function is even or odd, we need to evaluate f(-x).

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

Comparing f(x) and f(-x), we can see that f(x) = -f(-x). Since they are equal up to a negative sign, this function is odd.

9. f(x) = -4x^5 + x^3

To determine whether this function is even or odd, we need to evaluate f(-x).

f(-x) = -4(-x)^5 + (-x)^3
= -4(-x^5) + (-x^3)
= 4x^5 - x^3

Comparing f(x) and f(-x), we can see that f(x) = f(-x). They are equal without any difference in sign. Therefore, this function is even.

In summary:
- Function f(x) = x^3 - x^2 is odd.
- Function f(x) = -4x^5 + x^3 is even.

To determine whether a function is even, odd, or neither, you can use the properties of even and odd functions.

An even function is symmetric about the y-axis, meaning that if you reflect the graph of the function across the y-axis, it remains unchanged. Mathematically, for an even function f(x), it satisfies the condition f(-x) = f(x).

An odd function is symmetric about the origin, meaning that if you reflect the graph of the function across the origin, it remains unchanged. Mathematically, for an odd function f(x), it satisfies the condition f(-x) = -f(x).

Here's how you can determine whether the given functions are even, odd, or neither:

8. f(x) = x^3 - x^2
To check if the function is even or odd, let's evaluate f(-x) and compare it to f(x):
f(-x) = (-x)^3 - (-x)^2 = -x^3 - x^2
Now, compare f(-x) and f(x):
f(-x) = -x^3 - x^2
f(x) = x^3 - x^2
Since f(-x) = f(x), the function is even.

9. f(x) = -4x^5 + x^3
To check if the function is even or odd, let's evaluate f(-x) and compare it to f(x):
f(-x) = -4(-x)^5 + (-x)^3 = 4x^5 - x^3
Now, compare f(-x) and f(x):
f(-x) = 4x^5 - x^3
f(x) = -4x^5 + x^3
Since f(-x) = f(x), the function is even.

Therefore, for problem 8, the function f(x) = x^3 - x^2 is even, and for problem 9, the function f(x) = -4x^5 + x^3 is also even.