Match each function to its graph based on symmetry properties.

Problem Overview:

The task involves matching functions from the top row (A–M) with their corresponding graphs in the bottom row (N–Y). The functions are categorized into three types:
1. Symmetric about the origin (functions where \( f(-x) = -f(x) \)).
2. Symmetric about the y-axis (functions where \( f(-x) = f(x) \)).
3. No symmetry (functions that do not satisfy either of the above conditions).

We will analyze each function, determine its symmetry properties, and match it with the appropriate graph.

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Step-by-Step Analysis:

#### Functions and Their Symmetry:

1. A. \( f(-x) = -f(x) \)
- This indicates the function is odd and symmetric about the origin.
- Examples: \( f(x) = x^3 \), \( f(x) = \sin(x) \).

2. B. \( f(x) - f(-x) \)
- This expression does not directly indicate symmetry. We need to evaluate specific functions to determine symmetry.

3. C. \( f(x) \cdot f(-x) \)
- This expression does not directly indicate symmetry. We need to evaluate specific functions to determine symmetry.

4. D. \( f(x) = 2x^3 - 4x \)
- Check for odd symmetry:
\[
f(-x) = 2(-x)^3 - 4(-x) = -2x^3 + 4x = -(2x^3 - 4x) = -f(x)
\]
- This function is odd and symmetric about the origin.

5. E. \( f(x) = 2x^4 + 2x^2 - 7 \)
- Check for even symmetry:
\[
f(-x) = 2(-x)^4 + 2(-x)^2 - 7 = 2x^4 + 2x^2 - 7 = f(x)
\]
- This function is even and symmetric about the y-axis.

6. F. \( f(x) = x^3 + 4 \)
- Check for odd or even symmetry:
\[
f(-x) = (-x)^3 + 4 = -x^3 + 4
\]
- \( f(-x) \neq f(x) \) and \( f(-x) \neq -f(x) \).
- This function has no symmetry.

7. G. \( f(x) = \sqrt{2x} - 1 \)
- The domain is \( x \geq 0 \), so the function is not defined for negative \( x \).
- This function has no symmetry.

8. H. \( f(x) = x^4 \)
- Check for even symmetry:
\[
f(-x) = (-x)^4 = x^4 = f(x)
\]
- This function is even and symmetric about the y-axis.

9. I. Symmetric about the origin
- This is a general description of odd functions.

10. J. Symmetric about the y-axis
- This is a general description of even functions.

11. K. No symmetry
- This is a general description for functions that are neither odd nor even.

---

#### Graph Analysis:

Now, we analyze the graphs (N–Y) to determine their symmetry properties:

- N: Parabola opening upwards, symmetric about the y-axis.
- Symmetry: Even (about the y-axis).

- O: Cubic curve passing through the origin, symmetric about the origin.
- Symmetry: Odd (about the origin).

- P: Cubic curve shifted vertically, not symmetric about the origin or y-axis.
- Symmetry: None.

- Q: Linear function with a positive slope, not symmetric about the origin or y-axis.
- Symmetry: None.

- R: Square root function, defined only for \( x \geq 0 \), not symmetric.
- Symmetry: None.

- S: Parabola opening downwards, symmetric about the y-axis.
- Symmetry: Even (about the y-axis).

- T: Cubic curve shifted vertically, not symmetric about the origin or y-axis.
- Symmetry: None.

- U: Cubic curve passing through the origin, symmetric about the origin.
- Symmetry: Odd (about the origin).

---

#### Matching Functions to Graphs:

1. A. \( f(-x) = -f(x) \) → Odd symmetry → Match with O or U.
2. B. \( f(x) - f(-x) \) → Not directly indicative of symmetry → Evaluate specific functions.
3. C. \( f(x) \cdot f(-x) \) → Not directly indicative of symmetry → Evaluate specific functions.
4. D. \( f(x) = 2x^3 - 4x \) → Odd symmetry → Match with O or U.
5. E. \( f(x) = 2x^4 + 2x^2 - 7 \) → Even symmetry → Match with N or S.
6. F. \( f(x) = x^3 + 4 \) → No symmetry → Match with P, Q, R, or T.
7. G. \( f(x) = \sqrt{2x} - 1 \) → No symmetry → Match with P, Q, R, or T.
8. H. \( f(x) = x^4 \) → Even symmetry → Match with N or S.
9. I. Symmetric about the origin → Odd symmetry → Match with O or U.
10. J. Symmetric about the y-axis → Even symmetry → Match with N or S.
11. K. No symmetry → Match with P, Q, R, or T.

---

#### Final Matches:

- A: O (or U, but U is better as it matches \( f(x) = 2x^3 - 4x \)).
- B: P (evaluate specific functions).
- C: Q (evaluate specific functions).
- D: O.
- E: N.
- F: P.
- G: R.
- H: S.
- I: U.
- J: S.
- K: T.

---

Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{A} \to \text{O}, & \text{B} \to \text{P}, \\
\text{C} \to \text{Q}, & \text{D} \to \text{O}, \\
\text{E} \to \text{N}, & \text{F} \to \text{P}, \\
\text{G} \to \text{R}, & \text{H} \to \text{S}, \\
\text{I} \to \text{U}, & \text{J} \to \text{S}, \\
\text{K} \to \text{T}. &
\end{array}
}
\]