Question

The function \( f(x) = \frac{\sqrt{3x^2 - 5x - 2}{2x^2 - 7x + 5} \) has discontinuous points at \( x = \dots \)}

• A. \( \frac{5}{2}, 2 \)
• B. \( -\frac{1}{3}, 2 \)
• C. \( 1, \frac{5}{2} \)
• D. \( -\frac{1}{3}, 1 \)

Hint:

For rational functions with square roots, check both the denominator (for division by zero) and the expression inside the square root (for non-negative values).

Answer 1

The Correct Option is A

For the function \( f(x) = \frac{\sqrt{3x^2 - 5x - 2}}{2x^2 - 7x + 5} \), we need to find the values of \( x \) where the function is undefined or discontinuous. Step 1: The function will be undefined if the denominator \( 2x^2 - 7x + 5 = 0 \). Solving for \( x \): \[ 2x^2 - 7x + 5 = 0 \] Using the quadratic formula: \[ x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(2)(5)}}{2(2)} = \frac{7 \pm \sqrt{49 - 40}}{4} = \frac{7 \pm \sqrt{9}}{4} \] \[ x = \frac{7 \pm 3}{4} \quad \Rightarrow \quad x = \frac{10}{4} = \frac{5}{2} \quad \text{or} \quad x = \frac{4}{4} = 1 \] Step 2: The function will also be discontinuous if the expression inside the square root is negative, i.e., \( 3x^2 - 5x - 2 \geq 0 \). After solving the inequalities and considering the points where the function becomes undefined, we find that the function is discontinuous at \( x = \frac{5}{2} \) and \( x = 2 \). % Final Answer The discontinuous points are \( \frac{5}{2} \) and \( 2 \).