Sure, let's simplify the explanation and approach for grade 10 and 11 levels.

Question 9: Simplify the expression $( - 3a^{-2}b^2)^{-3} \cdot ( - a^{-3}b^4)$

1. Simplify each term separately: $( - 3a^{-2}b^2)^{-3} = (-3)^{-3} \cdot (a^{-2})^{-3} \cdot (b^2)^{-3}$ $= -\frac{1}{27} \cdot a^{6} \cdot b^{-6}$ $= -\frac{a^6}{27b^6}$

   For the second term: $( - a^{-3}b^4) = -a^{-3}b^4$

2. Combine the two terms: $-\frac{a^6}{27b^6} \cdot (-a^{-3}b^4) = \frac{a^6 \cdot a^{-3}}{27b^6} \cdot b^4$ $= \frac{a^{6-3} \cdot b^{4-6}}{27} = \frac{a^3 b^{-2}}{27}$ $= \frac{a^3}{27b^2}$

Therefore, the simplified expression is: $\boxed{\frac{a^3}{27b^2}}$

Question 10: Analyze the function $f(x) = -3\left( \frac{1}{2} \right)^x + 2$

a) Domain and Range:

• Domain: The domain of $f(x)$ is all real numbers $\mathbb{R}$, because exponential functions are defined for all real numbers.

• Range: The range is determined by analyzing the behavior of the function. Since $\left( \frac{1}{2} \right)^x$ is always positive and tends to 0 as $x \to \infty$ and grows very large as $x \to -\infty$, the function $-3\left( \frac{1}{2} \right)^x$ will range from $-3$ to $-\infty$. Adding 2 to this will range from $-1$ to $2$. Therefore, the range is: $(-1, 2]$

b) y-intercept:

The y-intercept occurs when $x = 0$: $f(0) = -3\left( \frac{1}{2} \right)^0 + 2 = -3(1) + 2 = -1$

Therefore, the y-intercept is $(0, -1)$.

c) Whether the function is increasing or decreasing:

• Increasing/Decreasing: The base $\frac{1}{2}$ is less than 1, indicating the exponential function $\left( \frac{1}{2} \right)^x$ is decreasing. Multiplying by $-3$ and adding 2 maintains a decreasing behavior.
• Rate: The rate at which the function decreases depends on the exponential rate of $\left( \frac{1}{2} \right)^x$, which decreases exponentially.

Thus, the function is decreasing at an exponentially decreasing rate.

Question 11: Determine the equation of the exponential function with base 4

Given:

• Reflected in the x-axis
• Asymptote at y = 3
• Passes through point (1, -5)

Let's start by writing the general form of an exponential function with base 4: $f(x) = a \cdot 4^x + b$

Reflecting in the x-axis: $f(x) = -a \cdot 4^x + b$

Since the asymptote is y = 3, $b = 3$: $f(x) = -a \cdot 4^x + 3$

It passes through (1, -5), so: $-5 = -a \cdot 4^1 + 3$ $-5 = -4a + 3$ $-8 = -4a$ $a = 2$

Therefore, the function is: $f(x) = -2 \cdot 4^x + 3$

If you need further details or have any questions, feel free to ask! Here are 5 questions you might be interested in:

1. How do you solve logarithmic equations?
2. What is the derivative of an exponential function?
3. How do you find the inverse of a function?
4. What are the applications of exponential functions in real life?
5. How can you determine the concavity of a function?

Tip: When dealing with exponential functions, always pay attention to the base and transformations like reflections and translations to determine the function's behavior.