Consider the following functions:

f(x)=sin(x^4-x^2)
h(x)=(|x|-3)^3
g(x)=1n(|x|)+3
s(x)=sin^3(x)

Which of the following is true?

h and g are even, f and s are odd.
f is even, h and s are odd.
h and s are even, f is odd.
***f and h are even, s is odd.
f, h, and s are odd.

correct, though g is also even

So what’s the answer

Well, isn't this a fun party of functions! Let's break it down and find out what's true.

First up, let's talk about even and odd functions. An even function is symmetric about the y-axis, meaning f(x) = f(-x), while an odd function is symmetric about the origin, meaning f(x) = -f(-x).

Looking at f(x) = sin(x^4 - x^2), it's not symmetrical about the y-axis or the origin. So, it's safe to say that f(x) is not even or odd.

Moving on to h(x) = (|x| - 3)^3, if we replace x with -x, we get h(-x) = (|-x| - 3)^3. Since |-x| is the same as |x|, we can write this as h(-x) = (|x| - 3)^3, which is equal to h(x). So, h(x) is indeed even.

Lastly, let's talk about s(x) = sin³(x). If we replace x with -x, we get s(-x) = sin³(-x). Since sin(-x) = -sin(x), we can rewrite this as s(-x) = -sin³(x). As you can see, s(-x) is not equal to -s(x), so s(x) is not even. But, s(-x) is equal to -s(x), which means s(x) is odd.

So, in conclusion, f(x) and h(x) are even, while s(x) is odd. The correct option is f and h are even, s is odd. And remember, it's always good to keep the math party entertaining!

To determine whether a function is even or odd, we need to consider the function's symmetry.

1. An even function is symmetric with respect to the y-axis. This means that if you reflect the part of the graph to the right of the y-axis to the left, it will match. Mathematically, an even function satisfies f(x) = f(-x) for all x in its domain.

2. An odd function is symmetric with respect to the origin (0,0). This means that if you reflect the entire graph across the origin, it will match. Mathematically, an odd function satisfies f(x) = -f(-x) for all x in its domain.

Now let's analyze each function:

f(x) = sin(x^4 - x^2)
This function involves the composition of an odd function (sin) with an even function (x^4 - x^2). However, since sin is an odd function, the overall function f(x) will be odd. Therefore, f is odd.

h(x) = (|x| - 3)^3
This function involves the absolute value, which is an even function. Then raising it to the power of 3 does not change its symmetry. As a result, h(x) is even.

g(x) = ln(|x|) + 3
The natural logarithm function, ln(x), is undefined for x ≤ 0. However, for x > 0, it is an increasing function with respect to the y-axis, and therefore it is neither even nor odd. Since ln(|x|) does not have any specific symmetry, the entire function g(x) is neither even nor odd.

s(x) = sin^3(x)
Again, this function involves the composition of an odd function (sin) with an even power (3). Since sin is an odd function, the overall function s(x) will be odd. Therefore, s is odd.

After analyzing all the given functions, we can conclude that:
f and h are even,
s is odd.

Therefore, the correct answer is: "f and h are even, s is odd."