Question

The sum of the least positive integer and the greatest negative integer in the range of the function \( f(x) = \dfrac{x^2 - 5x + 7}{x^2 - 5x - 7} \) is:

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(-1\)

Hint:

To find the range of rational functions like \( \frac{P(x)}{Q(x)} \), convert the expression to an equation \( y = f(x) \), rearrange, and use discriminant conditions \( D \geq 0 \) for real solutions.

Answer 1

The Correct Option is B

We are given: \[ f(x) = \frac{x^2 - 5x + 7}{x^2 - 5x - 7} \] Let’s simplify by substituting \( y = f(x) \), and solve for \( y \): \[ y = \frac{x^2 - 5x + 7}{x^2 - 5x - 7} \Rightarrow y(x^2 - 5x - 7) = x^2 - 5x + 7 \] \[ yx^2 - 5yx - 7y = x^2 - 5x + 7 \] Bring all terms to one side: \[ yx^2 - x^2 - 5yx + 5x - 7y - 7 = 0 \Rightarrow (y - 1)x^2 - 5(y - 1)x - 7y - 7 = 0 \] This is a quadratic in \( x \), and for real \( x \), the discriminant \( D \geq 0 \): \[ D = [-5(y - 1)]^2 - 4(y - 1)(-7y - 7) \] \[ = 25(y - 1)^2 + 28(y - 1)(y + 1) \] \[ = 25(y^2 - 2y + 1) + 28(y^2 - 1) = 25y^2 - 50y + 25 + 28y^2 - 28 = 53y^2 - 50y - 3 \] We want: \[ 53y^2 - 50y - 3 \geq 0 \] Solve the inequality: Find the roots of the quadratic: \[ y = \frac{50 \pm \sqrt{(-50)^2 + 4 \cdot 53 \cdot 3}}{2 \cdot 53} = \frac{50 \pm \sqrt{2500 + 636}}{106} = \frac{50 \pm \sqrt{3136}}{106} = \frac{50 \pm 56}{106} \] So: \[ y = \frac{106}{106} = 1, \text{or} y = \frac{-6}{106} = -\frac{3}{53} \] Thus, the range of \( f(x) \) is: \[ (-\infty, -\frac{3}{53}] \cup [1, \infty) \] 
The greatest negative integer in the range is \( -1 \), and the least positive integer is \( 1 \). Their sum: \[ -1 + 1 = \boxed{0} \] Wait — but this contradicts the marked answer in the image. Let’s recheck carefully. Actually, we observe that \( -1 \notin (-\infty, -\frac{3}{53}] \), because \( -1 < -\frac{3}{53} \approx -0.0566 \) — so yes, \( -1 \in \) range. And \( 1 \in [1, \infty) \Rightarrow \) valid. So: \[ \text{Sum} = -1 + 1 = \boxed{0} \] Hence, correct answer should be (1) 0 — not (2).
Correction Note: Based on discriminant analysis, the actual answer should be (A) \(0\). The originally marked answer (B) \(1\) appears to be incorrect.