To find the domain of the function
\[
f(x) = \frac{\log_4(5 - [x - 1] - [x]^2)}{x^2 + x - 2}
\]
where \([x]\) denotes the greatest integer function, we need to consider two main conditions:
1. The argument of the logarithm must be greater than 0.
2. The denominator must not be equal to 0.
### Step 1: Analyze the logarithmic function
The logarithmic function \(\log_4(5 - [x - 1] - [x]^2)\) is defined when:
\[
5 - [x - 1] - [x]^2 > 0
\]
Rearranging this gives:
\[
5 > [x - 1] + [x]^2
\]
### Step 2: Rewrite the greatest integer function
Using the property of the greatest integer function, we can express \([x - 1]\) as \([x] - 1\). Therefore, we have:
\[
5 > ([x] - 1) + [x]^2
\]
This simplifies to:
\[
5 > [x]^2 + [x] - 1
\]
or
\[
[x]^2 + [x] - 6 < 0
\]
### Step 3: Factor the quadratic inequality
Now we can factor the quadratic:
\[
[x]^2 + [x] - 6 = ([x] - 2)([x] + 3) < 0
\]
### Step 4: Determine the intervals
To solve the inequality \(([x] - 2)([x] + 3) < 0\), we find the roots:
- The roots are \([-3, 2]\).
The quadratic is negative between the roots, so we have:
\[
-3 < [x] < 2
\]
### Step 5: Interpret the greatest integer function
The greatest integer \([x]\) can take integer values. Thus, the possible integer values for \([x]\) are:
\[
-3, -2, -1, 0, 1
\]
This implies:
\[
-3 \leq x < 2
\]
### Step 6: Analyze the denominator
Next, we need to ensure the denominator \(x^2 + x - 2\) is not equal to 0. We can factor this as:
\[
(x - 1)(x + 2) \neq 0
\]
This gives us the restrictions:
\[
x \neq 1 \quad \text{and} \quad x \neq -2
\]
### Step 7: Combine the conditions
Now we combine the conditions from the logarithm and the denominator:
1. From the logarithm, we have \(-3 \leq x < 2\).
2. Exclude \(x = 1\) and \(x = -2\).
Thus, the domain of \(f(x)\) is:
\[
[-3, -2) \cup (-2, 1) \cup (1, 2)
\]
### Final Domain
The final domain of the function \(f(x)\) is:
\[
[-3, -2) \cup (-2, 1) \cup (1, 2)
\]
