Question

If \( f : \mathbb{R} \to \mathbb{R} \) and \( g : \mathbb{R} \to \mathbb{R} \) are two functions defined by \( f(x) = 2x - 3 \) and \( g(x) = 5x^2 - 2 \), then the least value of the function \((g \circ f)(x)\) is:

• A. \(-2\)
• B. \(2\)
• C. \(-4\)
• D. \(4\)

Hint:

For composite functions \( (g \circ f)(x) = g(f(x)) \), substitute \( f(x) \) into \( g \), then find minima/maxima by standard calculus or quadratic vertex formula.

Answer 1

The Correct Option is A

Step 1: Define the composite function
\[ (g \circ f)(x) = g(f(x)) = g(2x-3) = 5(2x-3)^2 - 2 \] Step 2: Expand and simplify
\[ (g \circ f)(x) = 5(4x^2 - 12x + 9) - 2 = 20x^2 - 60x + 45 - 2 = 20x^2 - 60x + 43 \] Step 3: Find the minimum value of the quadratic
Since the coefficient of \(x^2\) is positive (20), the parabola opens upward and has a minimum at \[ x = -\frac{b}{2a} = \frac{60}{2 \times 20} = \frac{60}{40} = \frac{3}{2} \] Step 4: Calculate minimum value
\[ (g \circ f)(\frac{3}{2}) = 20 \left(\frac{3}{2}\right)^2 - 60 \left(\frac{3}{2}\right) + 43 = 20 \times \frac{9}{4} - 90 + 43 = 45 - 90 + 43 = -2 \]