The problem is asking about composite and inverse functions, and also whether the functions $f(x)$ and $g(x)$ are even or odd.

Here's the breakdown of the problem:

1. Given Functions:

   • $f(x) = x^4 - 3x^2 + 2$
   • $g(x) = \sin(\pi x)$

   Tasks:

   • Determine whether each function is even or odd.
   • Find the composite functions $f \circ g$, $g \circ f$, and $f \circ f$.

Step 1: Check for Even or Odd Functions

• A function $h(x)$ is even if $h(-x) = h(x)$ for all $x$.
• A function $h(x)$ is odd if $h(-x) = -h(x)$ for all $x$.

For $f(x) = x^4 - 3x^2 + 2$:

• Let's compute $f(-x)$: $f(-x) = (-x)^4 - 3(-x)^2 + 2 = x^4 - 3x^2 + 2 = f(x)$ Since $f(-x) = f(x)$, $f(x)$ is an even function.

For $g(x) = \sin(\pi x)$:

• Let's compute $g(-x)$: $g(-x) = \sin(\pi(-x)) = \sin(-\pi x) = -\sin(\pi x) = -g(x)$ Since $g(-x) = -g(x)$, $g(x)$ is an odd function.

Step 2: Find Composite Functions

• $f \circ g$: This is $f(g(x))$, meaning you substitute $g(x)$ into $f(x)$: $f(g(x)) = f(\sin(\pi x)) = (\sin(\pi x))^4 - 3(\sin(\pi x))^2 + 2$ Simplified: $f(g(x)) = \sin^4(\pi x) - 3\sin^2(\pi x) + 2$

• $g \circ f$: This is $g(f(x))$, meaning you substitute $f(x)$ into $g(x)$: $g(f(x)) = \sin(\pi f(x)) = \sin(\pi(x^4 - 3x^2 + 2))$ This expression can't be simplified much further.

• $f \circ f$: This is $f(f(x))$, meaning you substitute $f(x)$ into $f(x)$: $f(f(x)) = f(x^4 - 3x^2 + 2) = (x^4 - 3x^2 + 2)^4 - 3(x^4 - 3x^2 + 2)^2 + 2$ Expanding this would lead to a large polynomial, but this is the basic form.

Let me know if you'd like a more detailed explanation of any part or if you'd like to explore specific examples with values.

Relative Questions:

1. How do you determine if a function is neither even nor odd?
2. What is the significance of composite functions in mathematical analysis?
3. How do inverse functions differ from composite functions?
4. Can you think of real-world applications of even and odd functions?
5. How do the properties of $\sin(x)$ and polynomials affect the composite functions in this problem?

Tip: When working with composite functions, it can be useful to focus on the domain of each function to ensure the composition is valid throughout.