The problem asks to find the set of values of the function:

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

To find the set of possible values of $f(x)$, we need to explore the behavior of the individual components of the function:

1. $\cos^2(x)$ is always between 0 and 1, since $\cos(x)$ ranges between -1 and 1. Therefore, $\cos^2(x)$ is always in the interval $[0, 1]$.

2. $\sqrt{2} \sin \left( 2x - \frac{\pi}{4} \right)$ will vary because the sine function ranges between -1 and 1. Thus, $\sqrt{2} \sin \left( 2x - \frac{\pi}{4} \right)$ will range from $-\sqrt{2}$ to $\sqrt{2}$.

Step 1: Analyzing the range of the sum

The total range of the function is determined by adding the two parts together:

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

• The term $\cos^2(x)$ contributes between 0 and 1.
• The term $\sqrt{2} \sin \left( 2x - \frac{\pi}{4} \right)$ contributes between $-\sqrt{2}$ and $\sqrt{2}$.

So, the set of possible values of $f(x)$ will be the sum of these two ranges. The lowest value occurs when $\cos^2(x) = 0$ and $\sqrt{2} \sin \left( 2x - \frac{\pi}{4} \right) = -\sqrt{2}$, giving:

$f(x) = 0 - \sqrt{2} = -\sqrt{2}$

The highest value occurs when $\cos^2(x) = 1$ and $\sqrt{2} \sin \left( 2x - \frac{\pi}{4} \right) = \sqrt{2}$, giving:

$f(x) = 1 + \sqrt{2}$

Thus, the set of values of $f(x)$ is:

$\left[ -\sqrt{2}, 1 + \sqrt{2} \right]$

Conclusion:

The set of values of the function is the interval:

$\boxed{ \left[ -\sqrt{2}, 1 + \sqrt{2} \right] }$

Let me know if you need further details or clarification!

Here are some related questions to explore:

1. How can we analyze a function that combines trigonometric and polynomial terms like this?
2. What are the properties of trigonometric functions that are relevant for finding function ranges?
3. How does the amplitude of the sine function affect the overall range of the sum?
4. How can you generalize the process of finding function values for other types of functions?
5. What is the relationship between the period of a trigonometric function and its range?

Tip: Always check the individual components of a function before combining them to determine the overall range!