Question

The domain of the function \(f(x) = e^{\sqrt{5x-3-2x^2}}\) is

• A. \((1,2)\)
• B. \((-1,\frac{3}{2})\)
• C. \([1,\frac{3}{2}]\)
• D. \((-1,0)\)

Hint:

The exponential is defined everywhere, so only the expression under the square root must be non-negative.

Answer 1

The Correct Option is C

Step 1: Find where the radicand is non-negative.
Graph $g(x) = -2x^2 + 5x - 3$, a downward parabola with roots at $x = 1$ and $x = 1.5$.

Step 2: Read the graph.
A downward parabola is above or on the axis between its roots, so $g(x) \geq 0$ for $1 \leq x \leq 1.5$.

Step 3: Check a value.
At $x = 1.2$: $6 - 3 - 2.88 = 0.12 > 0$ and at $x = 0$: $-3 < 0$. This is consistent.

Final Answer:
Option (C). \[ \boxed{\left[1, \frac{3}{2}\right]} \]