Let's analyze the questions and solutions based on the uploaded image:

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

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

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

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

Would you like further details or clarifications on any of these solutions?

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.