I can help you analyze and explain the math content from your uploaded image. Let me interpret and clarify the concepts.

Observations:

1. Functions and Even/Odd Analysis:
   • The given tasks involve checking whether the provided functions $f(x)$ are even, odd, or neither.
   • This is done by evaluating $f(-x)$ and comparing it to $f(x)$ and $-f(x)$.

Key Steps from Your Work:

1. First Function: $f(x) = \sqrt{x} - 4$

   • To test for evenness: Substitute $-x$ into $f(x)$.
     • $f(-x) = \sqrt{-x} - 4$.
     • Since $f(-x) \neq f(x)$, it is not even.
   • To test for oddness: Check if $f(-x) = -f(x)$.
     • $-f(x) = -( \sqrt{x} - 4 ) = -\sqrt{x} + 4$.
     • $f(-x) \neq -f(x)$, so it is not odd.
   • Conclusion: $f(x)$ is neither even nor odd.

2. Second Function: $f(x) = 2x^2 - 3x + 4$

   • Substitute $-x$ into $f(x)$:
     • $f(-x) = 2(-x)^2 - 3(-x) + 4 = 2x^2 + 3x + 4$.
   • Compare $f(-x)$ to $f(x)$:
     • $f(-x) \neq f(x)$, so it is not even.
   • Compare $f(-x)$ to $-f(x)$:
     • $-f(x) = -(2x^2 - 3x + 4) = -2x^2 + 3x - 4$.
     • $f(-x) \neq -f(x)$, so it is not odd.
   • Conclusion: $f(x)$ is neither even nor odd.

Do you need:

1. A detailed breakdown of the steps above?
2. Further examples of even/odd function analysis?
3. Additional graph interpretations for the provided functions?
4. Explanations for related concepts like symmetry in graphs?
5. Clarifications on why specific conditions fail in the tests?

Tip:

When testing functions for evenness or oddness:

• Even Function: $f(-x) = f(x)$ (Symmetric about the y-axis).
• Odd Function: $f(-x) = -f(x)$ (Symmetric about the origin).

Let me know how I can assist further!