https://file.aiquickdraw.com/mathbot/file/1726701608865ges3djlc.jpg

Let's break down the tasks from the uploaded image and analyze them one by one.

### 1. Task 1:
**(i)** Give an example of a mapping that is not a function. Then, for the function \( f : \mathbb{R} \to \mathbb{Z}^+ \) defined by:

\[
f(x) = \frac{(x^2 - 6)}{(x^2 + 6)^2}
\]

Prove that \( f(x) \) is one-to-one and onto.

**(ii)** State the necessity of L'Hôpital's Rule in calculus, with verification by an example using the basic definition.

### 2. Task 2:
**(i)** Discuss the continuity of the following piecewise function \( f(x) \):

\[
f(x) = 
\begin{cases} 
\frac{x^2 - x}{x^2 + 1} & -3 \leq x < -1 \\
2 & x = -1 \\
x^3 - 2x + 1 & -1 \leq x \leq 2
\end{cases}
\]

Determine whether the function is continuous for all points in \( x = [-1, 2] \).

**(ii)** Evaluate the limit \( \lim_{m \to 5} (10 + |m - 5|) \) and discuss if all operations (like +, -, etc.) are possible for two limits without any restriction.

### 3. Task 3:
**a)** Determine the domain and range of the following functions, and also sketch the graph:

- (i) \( f(x) = -\sqrt{x^2 + 3x - 10} \)
- (ii) \( g(x) = \frac{-1x + 51}{x + 5} \)

**b)** Find the differential coefficients of the following:

- (i) \( \tan^2(2\ln x) \)
- (ii) \( \sqrt{x} \sin^2(3x^3 - 5) \)

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Now, do you want to focus on solving one particular problem, or should I start explaining the entire breakdown of one section at a time? Feel free to ask about any part in more detail.

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Here are some related questions for exploration:
1. How do you determine if a function is "onto" and "one-to-one" in general?
2. Can L'Hôpital's Rule always be applied for indeterminate forms?
3. What are the basic conditions for a function to be continuous at a point?
4. How does one determine the domain and range of square root functions?
5. What are the standard rules for differentiating trigonometric functions involving logarithms?

**Tip:** When evaluating limits, always consider whether the function is continuous or if indeterminate forms like \(\frac{0}{0}\) occur, which may suggest the use of L'Hôpital's Rule.

The image contains mathematical tasks related to function analysis, continuity, domain, range, and differentiation. These tasks include proving a function is one-to-one and onto, using L'Hôpital's Rule, discussing continuity of a piecewise function, and determining domains, ranges, and differentiating specific expressions.

Explore problems related to function analysis, continuity for piecewise functions, L'Hôpital's Rule, and differentiation. This exercise covers key mathematical concepts such as proving one-to-one and onto functions, determining domains and ranges, and applying differentiation techniques. Suitable for Grades 11-12.

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