Find the range of the expression `y=(x^(2)-2x-8)/(x^(2)-4x-5)`, forall permissible value of x.

Answer

To find the range of the expression \( y = \frac{x^2 - 2x - 8}{x^2 - 4x - 5} \), we will follow these steps:

### Step 1: Identify the Domain
The first step is to determine the values of \( x \) for which the expression is defined. This means we need to find when the denominator is not equal to zero.

The denominator is:
\[ x^2 - 4x - 5 \]
We can factor this quadratic expression:
\[ x^2 - 4x - 5 = (x - 5)(x + 1) \]
Setting the denominator equal to zero gives us:
\[ (x - 5)(x + 1) = 0 \]
Thus, \( x = 5 \) and \( x = -1 \) are the points where the denominator is zero. Therefore, the expression is undefined at these points. The domain of \( y \) is all real numbers except \( x = 5 \) and \( x = -1 \).

### Step 2: Analyze the Expression
Next, we analyze the expression to find the range. We can rewrite \( y \) in terms of \( x \):
\[ y = \frac{x^2 - 2x - 8}{x^2 - 4x - 5} \]

### Step 3: Set Up for Range Determination
To find the range, we can manipulate the equation:
\[ y(x^2 - 4x - 5) = x^2 - 2x - 8 \]
This simplifies to:
\[ yx^2 - 4yx - 5y = x^2 - 2x - 8 \]
Rearranging gives us:
\[ (y - 1)x^2 + (2 - 4y)x + (5y - 8) = 0 \]
This is a quadratic equation in \( x \). For \( y \) to be in the range of the function, this quadratic must have real solutions. Therefore, the discriminant must be non-negative.

### Step 4: Calculate the Discriminant
The discriminant \( D \) of the quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:
\[ D = B^2 - 4AC \]
Here, \( A = y - 1 \), \( B = 2 - 4y \), and \( C = 5y - 8 \). Thus, the discriminant becomes:
\[ D = (2 - 4y)^2 - 4(y - 1)(5y - 8) \]

### Step 5: Simplify the Discriminant
Calculating \( D \):
\[ D = (2 - 4y)^2 - 4(y - 1)(5y - 8) \]
Expanding \( D \):
\[ D = (4 - 16y + 16y^2) - 4(5y^2 - 8y - 5y + 8) \]
\[ D = 4 - 16y + 16y^2 - 4(5y^2 - 13y + 8) \]
\[ D = 4 - 16y + 16y^2 - 20y^2 + 52y - 32 \]
\[ D = -4y^2 + 36y - 28 \]

### Step 6: Set the Discriminant Greater Than or Equal to Zero
To find the range of \( y \), we set the discriminant \( D \geq 0 \):
\[ -4y^2 + 36y - 28 \geq 0 \]
Dividing through by -4 (which reverses the inequality):
\[ y^2 - 9y + 7 \leq 0 \]

### Step 7: Factor the Quadratic
Factoring gives:
\[ (y - 7)(y - 2) \leq 0 \]
The critical points are \( y = 2 \) and \( y = 7 \).

### Step 8: Determine the Intervals
Using a sign chart or testing intervals, we find that the inequality holds true for:
\[ 2 \leq y \leq 7 \]

### Conclusion
Thus, the range of the expression \( y = \frac{x^2 - 2x - 8}{x^2 - 4x - 5} \) is:
\[ \boxed{[2, 7]} \]