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Let's go through the questions on your worksheet one by one.

### Question 1
**Express \( \sqrt{2 - 4x} \) in the form \( a + bx \) and state the condition under which the radical exists.**

To simplify \( \sqrt{2 - 4x} \), we need to ensure that the expression inside the square root is non-negative:

\[ 2 - 4x \geq 0 \]
\[ 2 \geq 4x \]
\[ \frac{1}{2} \geq x \]
\[ x \leq \frac{1}{2} \]

So, \( \sqrt{2 - 4x} \) is defined for \( x \leq \frac{1}{2} \).

### Question 2
**Write the exponential equation \( 3^y = x \) in logarithmic form.**

To convert the exponential equation \( 3^y = x \) to logarithmic form, we use the definition of a logarithm:

\[ y = \log_3(x) \]

### Question 3
**Find the exact value of \( \sin^{-1}(\frac{1}{2}) \).**

The inverse sine function \( \sin^{-1} \) (or arcsin) of \( \frac{1}{2} \) is the angle whose sine is \( \frac{1}{2} \). We know from trigonometry that:

\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

Therefore:

\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]

### Question 4
**Determine the side length \( x \) of the right triangle.**

The right triangle has sides of lengths \( x \), 3, and 5. Using the Pythagorean theorem:

\[ x^2 + 3^2 = 5^2 \]
\[ x^2 + 9 = 25 \]
\[ x^2 = 16 \]
\[ x = 4 \]

So, the side length \( x \) is 4.

### Summary of Answers:
1. \( \sqrt{2 - 4x} \) in the form \( a + bx \) is \( \sqrt{2 - 4x} \) with the condition \( x \leq \frac{1}{2} \).
2. \( 3^y = x \) in logarithmic form is \( y = \log_3(x) \).
3. \( \sin^{-1}(\frac{1}{2}) = \frac{\pi}{6} \).
4. The side length \( x \) of the triangle is 4.

Would you like more details on any of these solutions or have any other questions?

### Additional Questions:
1. What is the general form of a logarithmic function?
2. How do you derive the inverse trigonometric functions?
3. Can you convert \( \log_b(a) = c \) to its exponential form?
4. What are the domain and range of the inverse sine function?
5. How do you find the domain of a radical function?
6. What other methods can be used to simplify square root expressions?
7. How is the Pythagorean theorem applied in different geometrical contexts?
8. Can you solve for \( x \) if the right triangle's hypotenuse and one leg are given?

### Tip:
When dealing with logarithms, always ensure the base is positive and not equal to 1, and the argument is positive to maintain a valid logarithmic function.

Explore comprehensive solutions to mathematical problems involving square roots, exponential equations, inverse trigonometric functions, and the Pythagorean theorem. Understand the conditions and formulas necessary to solve these problems effectively. Suitable for Grades 9-12.

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