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### 18) \( f(x) = 6x^3 - 9 \)

- To determine if a function is even, odd, or neither:
  - A function is **even** if \( f(-x) = f(x) \).
  - A function is **odd** if \( f(-x) = -f(x) \).
  - If neither condition holds, the function is neither even nor odd.
  
For this function \( f(x) = 6x^3 - 9 \):
- Calculate \( f(-x) \):
  \[
  f(-x) = 6(-x)^3 - 9 = -6x^3 - 9
  \]
  \( f(-x) \neq f(x) \) and \( f(-x) \neq -f(x) \).

Thus, **the function is neither even nor odd**.
- Correct answer: **C) neither**

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### 19) \( f(x) = -8x^2 + 3 \)

- For even/odd determination:
  \[
  f(-x) = -8(-x)^2 + 3 = -8x^2 + 3 = f(x)
  \]
  Therefore, the function is **even**.
- Correct answer: **A) even**

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### 20) Graph: Determine if the function is increasing, decreasing, or constant on the interval \( (0, 1) \).

- The graph shows the function starts from \( y = 5 \) and decreases in the interval \( (0, 1) \).
- Correct answer: **C) decreasing**

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### 21) Graph: Determine if the function is increasing, decreasing, or constant on the interval \( (0, 3) \).

- The graph indicates the function is **constant** from \( x = 0 \) to \( x = 3 \), as the graph remains flat at \( y = 10 \).
- Correct answer: **B) constant**

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### Here are some related questions for practice:
1. How can you tell if a quadratic function is even or odd based on its equation?
2. For the cubic function \( f(x) = ax^3 + bx^2 + cx + d \), how can we determine whether it is increasing or decreasing over an interval?
3. What properties must a function's graph have to be classified as increasing or decreasing over a certain interval?
4. How does symmetry in a graph help identify whether a function is even or odd?
5. Why can a polynomial of odd degree never be an even function?

#### Tip:
Remember, to determine if a function is increasing or decreasing on an interval, focus on the slope or the direction the graph is moving. If the slope is positive, the function is increasing; if it's negative, the function is decreasing.

Given the functions f(x) = 6x^3 - 9 and f(x) = -8x^2 + 3, determine if they are even, odd, or neither. Additionally, determine whether the given graphs are increasing, decreasing, or constant on specified intervals.

Learn how to determine whether a function is even, odd, or neither using algebraic methods, and analyze graph behavior over specific intervals. This problem covers key concepts in function symmetry and graph analysis for high school students.

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