https://file.aiquickdraw.com/mathbot/file/1726540020423qu3ay3n0.jpg

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 \( \frac{3}{1+i} \times \frac{1-i}{1-i} = \frac{3(1-i)}{2} = \frac{3}{2} - \frac{3}{2}i \).

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### 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.

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### 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.

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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).

State whether TRUE or FALSE:
a) e^{\ln 3x} = 3x
b) Every natural number is a whole number
c) 5/(x+3) + 2x/(x-3) = 36/(x^2 - 9) has only one solution, which is x = -17/2
d) 8\sqrt{2} + 5\sqrt{98} - \sqrt{32} = 39\sqrt{2}
e) (-∞, 2] ∪ (-3, ∞) = (-∞, ∞)
f) x^2 - 7x + 3 = 0 has complex roots
g) f(x) = -g(x - 2) + 3 means that graph of g(x) is reflected about y-axis, shifted right 3 units, and upwards 3 units
h) 3/(1 + i) = 3/2 + (3/2)i
How many different arrangements of the word _____ are possible?
Identify vertical asymptotes, horizontal asymptotes, domain, and range, then sketch the graph.

Solve a quiz with questions on exponential functions, set theory, quadratic equations, complex numbers, and asymptotes. Ideal for high school students (Grades 10-12), with detailed solutions covering key algebraic and function transformations.

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