Let's break down the questions from the image and analyze them one by one:

1. State whether TRUE or FALSE:

a) $e^{\ln 3x} = 3x$

• True. Since $e^{\ln a} = a$, $e^{\ln 3x} = 3x$.

b) Every natural number is a whole number.

• True. Natural numbers are the set $\{1, 2, 3, \dots \}$ which are included in the whole numbers $\{0, 1, 2, 3, \dots \}$.

c) $\frac{5}{x+3} + \frac{2x}{x-3} = \frac{36}{x^2-9}$ has only one solution, which is $x = -\frac{17}{2}$.

• False. Solving this equation involves factoring and simplifying. $x = -\frac{17}{2}$ does not satisfy this equation.

d) $8\sqrt{2} + 5\sqrt{98} - \sqrt{32} = 39\sqrt{2}$

• False. Simplifying each term gives $8\sqrt{2} + 35\sqrt{2} - 4\sqrt{2} = 39\sqrt{2}$, which is not true.

e) $(-\infty, 2] \cup (-3, \infty) = (-\infty, \infty)$

• True. This union covers all real numbers.

f) $x^2 - 7x + 3 = 0$ has complex roots.

• True. The discriminant $\Delta = b^2 - 4ac = (-7)^2 - 4(1)(3) = 49 - 12 = 37$, which is positive, meaning the roots are real but irrational.

g) $f(x) = -g(x-2) + 3$ means that the graph of $g(x)$ is reflected about the x-axis, shifted 3 units right, and upwards 3 units.

• True. The transformation rules indicate this.

h) $\frac{3}{1+i} = \frac{3}{2} + \frac{3}{2}i$

• False. The correct simplified form of $\frac{3}{1+i}$ is $\frac{3}{1+i} \times \frac{1-i}{1-i} = \frac{3(1-i)}{2} = \frac{3}{2} - \frac{3}{2}i$.

2. How many different arrangements of the word (blank) are possible?

• This question is incomplete; it depends on the word provided. The number of arrangements is calculated as $n!$, where $n$ is the number of letters in the word.

3. Identify the vertical asymptotes, horizontal asymptotes, domain, and range. Then, sketch the graph.

• The function to analyze isn't given in the image, but if provided, we could solve for the asymptotes, domain, and range.

Let me know if you'd like more details or explanations for any specific parts.

Here are 5 related questions to explore further:

1. How do you solve logarithmic equations involving exponentials like $e^{\ln x}$?
2. What is the general rule for transforming functions on the graph (e.g., reflections, shifts)?
3. How can you find the complex roots of a quadratic equation using the discriminant?
4. How do unions of intervals work in set theory?
5. How do you simplify expressions involving square roots?

Tip: For solving fractional equations like part c), always check for restrictions on the variables (e.g., denominators cannot be zero).