Question

The range of the function \[ f(x)=\frac{x}{x^2-5x+9} \] is:

• A. \[ \left[\frac{1}{11},1\right] \]
• B. \[ \left[-\frac{1}{11},1\right] \]
• C. \[ \left[-1,-\frac{1}{11}\right] \]
• D. \[ \left[-1,\frac{1}{11}\right] \]

Hint:

To find the range of a rational function, put \(y=f(x)\), rearrange into a quadratic in \(x\), and use the condition \(\Delta \geq 0\) for real solutions.

Answer 1

The Correct Option is B

Step 1: Set up the equation for the range.
Let y = x/(x² - 5x + 9). Cross-multiplying yields yx² - (5y + 1)x + 9y = 0, which is a quadratic in x.

Step 2: Impose the reality condition on x.
For real x to exist, the discriminant must be non-negative: (5y + 1)² - 4(y)(9y) ≥ 0 → 25y² + 10y + 1 - 36y² ≥ 0 → -11y² + 10y + 1 ≥ 0. Multiplying by -1 reverses the inequality: 11y² - 10y - 1 ≤ 0.

Step 3: Solve the quadratic inequality.
Factorizing: (11y + 1)(y - 1) ≤ 0. The solution interval is -1/11 ≤ y ≤ 1.

Step 4: Confirm the endpoints are attainable.
At y = 1, the quadratic becomes x² - 6x + 9 = (x - 3)² = 0, yielding a real x. At y = -1/11, the discriminant vanishes, also producing a real x. Both endpoints belong to the range.

Step 5: Final conclusion.
The range of the function is [-1/11, 1].