I see the mathematical questions you've uploaded. Let me solve them step by step.

Question 9: Domain and Range of $f(x) = 2\sqrt{x+1} + 2$

To determine the domain:

• The square root function $\sqrt{x+1}$ is only defined for $x+1 \geq 0$, which means $x \geq -1$.

To determine the range:

• $\sqrt{x+1} \geq 0$, so $2\sqrt{x+1} \geq 0$.
• Adding 2 to this gives $f(x) \geq 2$.

Thus, the domain is $\{x \geq -1\}$, and the range is $\{y \geq 2\}$.
Correct Answer: Option (c).

Question 10: Restrictions on the Expression $\frac{x+5}{x-5}$

The denominator cannot be zero, so:

• $x-5 \neq 0$, which means $x \neq 5$.

Thus, the restriction is $x \neq 5$.
Correct Answer: Option (B).

Summary of Solutions:

• Q9 Answer: (c).
• Q10 Answer: (B).

Would you like me to explain any part in more detail or verify further? 😊

Here are 5 related questions to expand your understanding:

1. How do you determine the domain of a square root function in general?
2. Why does a fraction become undefined if its denominator is zero?
3. What would the range of $f(x) = 3\sqrt{x+2} + 1$ be?
4. Can you describe how transformations affect the domain and range of $\sqrt{x}$?
5. If $f(x) = \frac{x^2 - 25}{x - 5}$, what is the domain of this function?

Tip: Always check the denominator for restrictions and solve inequalities carefully for square roots!