Let's go through each of the questions on the uploaded worksheet and solve them properly:

Q1: What is the Natural Domain of the function:

$f(x) = \ln(x+1) - \sqrt{2 - x}$

The natural domain of a function is where the function is defined, considering the restrictions on the components.

• For the logarithmic part $\ln(x + 1)$, the argument $x + 1 > 0$, which means $x > -1$.
• For the square root part $\sqrt{2 - x}$, the argument inside the square root must be non-negative, so $2 - x \geq 0$, which simplifies to $x \leq 2$.

Thus, the domain is where both conditions hold: $-1 < x \leq 2$

The natural domain in set builder notation is:

$D = \{ x \in \mathbb{R} \mid -1 < x \leq 2 \}$

Q2: Find the inverse of the function $f(x) = \left(\frac{4}{x - 3}\right)^3$

To find the inverse, we follow these steps:

1. Start with the equation: $y = \left(\frac{4}{x - 3}\right)^3$

2. Take the cube root of both sides: $\sqrt[3]{y} = \frac{4}{x - 3}$

3. Solve for $x$: $x - 3 = \frac{4}{\sqrt[3]{y}}$

   $x = \frac{4}{\sqrt[3]{y}} + 3$

Thus, the inverse function is: $f^{-1}(x) = \frac{4}{\sqrt[3]{x}} + 3$

Q3: Define a function with the following natural domain:

$(-\infty, 5] \cup (6, \infty)$

You need to create a function whose domain excludes the intervals between $5$ and $6$. One way to approach this is by constructing a function that has restrictions based on division by zero or square roots.

For example, we could define:

$f(x) = \frac{1}{(x - 5)(x - 6)}$

This function is undefined at $x = 5$ and $x = 6$, which matches the required domain. Thus, the natural domain of this function would be:

$(-\infty, 5) \cup (6, \infty)$

Do you want details or have any questions?

Here are five related questions you might find helpful:

1. How can you find the range of a logarithmic function?
2. What are the steps to verify if two functions are inverses?
3. How do you handle inequalities involving square roots and logarithms?
4. What is the significance of domain restrictions in real-world applications of functions?
5. How can we graphically represent the natural domain of a function?

Tip: Always check the domain of the original function before finding its inverse, as not all functions have inverses over their entire domain!