To determine the symmetry of a function and classify it as even, odd, or neither, we analyze the behavior of $f(-x)$ and compare it with $f(x)$ and $-f(x)$. Let's evaluate each function:

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A. $f(x) = 7x^6 - 5x^2$

Step 1: Substitute $-x$ into the function

$f(-x) = 7(-x)^6 - 5(-x)^2 = 7x^6 - 5x^2$

Step 2: Compare $f(-x)$ with $f(x)$

Since $f(-x) = f(x)$, the function is even.

Step 3: Symmetry

An even function is symmetric about the y-axis.

Conclusion for A:

• Symmetry: y-axis
• Function type: even

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B. $f(x) = -3x^3 - 5x$

Step 1: Substitute $-x$ into the function

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

Step 2: Compare $f(-x)$ with $-f(x)$

The negative of the original function is: $-f(x) = -(-3x^3 - 5x) = 3x^3 + 5x$ Thus, $f(-x) = -f(x)$.

Step 3: Symmetry

If $f(-x) = -f(x)$, the function is odd and symmetric about the origin.

Conclusion for B:

• Symmetry: origin
• Function type: odd

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Final Results:

1. $f(x) = 7x^6 - 5x^2$:

   • Symmetry: y-axis
   • Type: even

2. $f(x) = -3x^3 - 5x$:

   • Symmetry: origin
   • Type: odd

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Would you like a detailed explanation of any step?
Here are 5 related questions to expand understanding:

1. How do you determine symmetry for polynomial functions?
2. Can a function have more than one type of symmetry?
3. What are real-world examples of even and odd functions?
4. How does symmetry affect the graph of a function?
5. Can symmetry help identify key properties of functions in calculus?

Tip: For polynomials, symmetry is often determined by the exponents. Terms with even exponents contribute to even symmetry, and terms with odd exponents contribute to odd symmetry.