The function provided is:

$f(x) = \log_2\left(\frac{\sin x - \cos x + \frac{3}{2}}{\sqrt{2}}\right)$

To find the range of this function, we need to determine the range of the expression inside the logarithm and ensure it is positive (since the logarithm is only defined for positive numbers).

Step-by-step Solution:

1. Simplify the Inside Expression:

   Let's consider the expression inside the logarithm:

   $y = \frac{\sin x - \cos x + \frac{3}{2}}{\sqrt{2}}$

2. Find the Range of $\sin x - \cos x$:

   Recall that the general form of $\sin x - \cos x$ can be rewritten using a phase shift:

   $\sin x - \cos x = \sqrt{2} \sin\left(x - \frac{\pi}{4}\right)$

   The range of $\sin\left(x - \frac{\pi}{4}\right)$ is $[-1, 1]$, so the range of $\sqrt{2} \sin\left(x - \frac{\pi}{4}\right)$ is $[- \sqrt{2}, \sqrt{2}]$.

3. Adjust the Range with the Constant:

   Now add $\frac{3}{2}$ to the range $[- \sqrt{2}, \sqrt{2}]$:

   $[-\sqrt{2}, \sqrt{2}] + \frac{3}{2} = \left(\frac{3}{2} - \sqrt{2}, \frac{3}{2} + \sqrt{2}\right)$

   The approximate range becomes:

   $\left(\frac{3}{2} - 1.414, \frac{3}{2} + 1.414\right) = (0.086, 2.914)$

4. Divide by $\sqrt{2}$:

   Now, divide this range by $\sqrt{2}$:

   $\left(\frac{0.086}{\sqrt{2}}, \frac{2.914}{\sqrt{2}}\right) \approx (0.061, 2.06)$

5. Determine the Range of $f(x)$:

   Since $f(x) = \log_2(y)$, where $y$ must be positive:

   $y \in (0.061, 2.06)$

   Therefore, the range of $f(x)$ is:

   $\left(\log_2(0.061), \log_2(2.06)\right)$

   Calculating these logarithms:

   $\log_2(0.061) \approx -4.03, \quad \log_2(2.06) \approx 1.05$

   Thus, the range of $f(x)$ is approximately:

   $(-4.03, 1.05)$

Conclusion:

The range of the function $f(x) = \log_2\left(\frac{\sin x - \cos x + \frac{3}{2}}{\sqrt{2}}\right)$ is approximately $(-4.03, 1.05)$.

Would you like more details on any part of the solution or have any questions?

Related Questions:

1. What is the domain of the function $f(x) = \log_2(x)$?
2. How do you transform trigonometric expressions into simpler forms?
3. What are the properties of logarithmic functions?
4. How does a phase shift affect the range of a sine or cosine function?
5. What is the significance of the base in logarithmic functions?

Tip:

When dealing with logarithmic functions, always ensure the argument is positive, as logarithms are only defined for positive real numbers.