Find the range of `f(x)=(x^2+34 x-71)/(x^2+2x-7)`

Explanation

To find the range of the function \( f(x) = \frac{x^2 + 34x - 71}{x^2 + 2x - 7} \), we can follow these steps:

### Step 1: Set the function equal to \( y \)
Assume \( f(x) = y \):
\[ y = \frac{x^2 + 34x - 71}{x^2 + 2x - 7} \]

### Step 2: Cross-multiply to eliminate the fraction
Rearranging gives:
\[ y(x^2 + 2x - 7) = x^2 + 34x - 71 \]
This leads to:
\[ yx^2 + 2yx - 7y = x^2 + 34x - 71 \]

### Step 3: Rearrange the equation
Rearranging the equation, we have:
\[ (y - 1)x^2 + (2y - 34)x + (7y - 71) = 0 \]

### Step 4: Analyze the discriminant
For \( x \) to have real values, the discriminant \( D \) of this quadratic must be greater than or equal to zero:
\[ D = (2y - 34)^2 - 4(y - 1)(7y - 71) \geq 0 \]

### Step 5: Expand the discriminant
Expanding the discriminant:
\[ D = (2y - 34)^2 - 4[(y - 1)(7y - 71)] \]
Calculating \( (2y - 34)^2 \):
\[ = 4y^2 - 136y + 1156 \]
Calculating \( 4(y - 1)(7y - 71) \):
\[ = 4(7y^2 - 71y - 7y + 71) = 4(7y^2 - 78y + 71) = 28y^2 - 312y + 284 \]
Now substituting back into the discriminant:
\[ D = 4y^2 - 136y + 1156 - (28y^2 - 312y + 284) \]
Combining like terms:
\[ D = 4y^2 - 28y^2 + 312y - 136y + 1156 - 284 \]
\[ D = -24y^2 + 176y + 872 \]

### Step 6: Set the discriminant greater than or equal to zero
Now we need to solve:
\[ -24y^2 + 176y + 872 \geq 0 \]
Dividing the entire inequality by -1 (which flips the inequality sign):
\[ 24y^2 - 176y - 872 \leq 0 \]

### Step 7: Find the roots of the quadratic
Using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 24, b = -176, c = -872 \):
\[ y = \frac{176 \pm \sqrt{(-176)^2 - 4 \cdot 24 \cdot (-872)}}{2 \cdot 24} \]
Calculating the discriminant:
\[ = 30976 + 83712 = 114488 \]
Now calculating the roots:
\[ y = \frac{176 \pm \sqrt{114488}}{48} \]

### Step 8: Approximate the roots
Calculating \( \sqrt{114488} \approx 338.5 \):
\[ y_1 = \frac{176 + 338.5}{48} \approx 10.7, \quad y_2 = \frac{176 - 338.5}{48} \approx -3.4 \]

### Step 9: Determine the intervals
The quadratic \( 24y^2 - 176y - 872 \) opens upwards (since the coefficient of \( y^2 \) is positive). Thus, the function is less than or equal to zero between the roots:
\[ y \in [-3.4, 10.7] \]

### Conclusion
The range of \( f(x) \) is:
\[ \text{Range of } f(x) = (-\infty, -3.4] \cup [10.7, \infty) \]