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.