Let `f(x) = cos^(2)x + cos^(2)(x + pi//3) + sin x sin (x + pi//3)` and
`g(x) ={{:(1, if x lt 5//4),(2, if 5//4 le x le 7//4),(3, if x gt 7//4):}`,
The number of elements in the range of `(gof)(x)` is …………….

Answer

To solve the problem, we need to analyze the functions \( f(x) \) and \( g(x) \) and find the number of elements in the range of the composite function \( g(f(x)) \).

### Step 1: Analyze the function \( f(x) \)

The function is given as:
\[ f(x) = \cos^2 x + \cos^2\left(x + \frac{\pi}{3}\right) + \sin x \sin\left(x + \frac{\pi}{3}\right) \]

Using the sine addition formula, we can simplify \( \sin\left(x + \frac{\pi}{3}\right) \):
\[ \sin\left(x + \frac{\pi}{3}\right) = \sin x \cos\frac{\pi}{3} + \cos x \sin\frac{\pi}{3} = \sin x \cdot \frac{1}{2} + \cos x \cdot \frac{\sqrt{3}}{2} \]

Now, substituting this back into \( f(x) \):
\[ f(x) = \cos^2 x + \cos^2\left(x + \frac{\pi}{3}\right) + \sin x \left(\frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x\right) \]

### Step 2: Simplify \( f(x) \)

Using the identity \( \cos^2\theta = 1 - \sin^2\theta \):
\[ \cos^2\left(x + \frac{\pi}{3}\right) = \cos^2 x \cos^2\frac{\pi}{3} - \sin^2 x \sin^2\frac{\pi}{3} = \cos^2 x \cdot \frac{1}{4} - \sin^2 x \cdot \frac{3}{4} \]

Thus,
\[ f(x) = \cos^2 x + \left(\frac{1}{4} \cos^2 x - \frac{3}{4} \sin^2 x\right) + \frac{1}{2} \sin^2 x + \frac{\sqrt{3}}{2} \sin x \cos x \]

Combining the terms gives:
\[ f(x) = \left(1 + \frac{1}{4}\right) \cos^2 x + \left(-\frac{3}{4} + \frac{1}{2}\right) \sin^2 x + \frac{\sqrt{3}}{2} \sin x \cos x \]
\[ = \frac{5}{4} \cos^2 x - \frac{1}{4} \sin^2 x + \frac{\sqrt{3}}{2} \sin x \cos x \]

### Step 3: Determine the range of \( f(x) \)

By analyzing \( f(x) \), we find that it can be expressed in terms of \( \sin^2 x \) and \( \cos^2 x \). The maximum and minimum values of \( f(x) \) can be evaluated, but we can also observe that \( f(x) \) is periodic and bounded.

### Step 4: Analyze the function \( g(x) \)

The function \( g(x) \) is defined as:
\[ g(x) = \begin{cases} 1 & \text{if } x < \frac{5}{4} \\ 2 & \text{if } \frac{5}{4} \leq x \leq \frac{7}{4} \\ 3 & \text{if } x > \frac{7}{4} \end{cases} \]

### Step 5: Find the range of \( g(f(x)) \)

Since \( f(x) \) is periodic and bounded, we need to determine where \( f(x) \) falls in relation to the critical points \( \frac{5}{4} \) and \( \frac{7}{4} \).

1. If \( f(x) < \frac{5}{4} \), then \( g(f(x)) = 1 \).
2. If \( \frac{5}{4} \leq f(x) \leq \frac{7}{4} \), then \( g(f(x)) = 2 \).
3. If \( f(x) > \frac{7}{4} \), then \( g(f(x)) = 3 \).

### Step 6: Conclusion

Since \( f(x) \) can take values that are either less than \( \frac{5}{4} \), between \( \frac{5}{4} \) and \( \frac{7}{4} \), or greater than \( \frac{7}{4} \), we can conclude that the range of \( g(f(x)) \) consists of the values \( 1, 2, \) and \( 3 \).

Thus, the number of elements in the range of \( g(f(x)) \) is:
\[ \boxed{3} \]