Question

If \( f : \mathbb{R} \to \mathbb{R}, g : \mathbb{R} \to \mathbb{R} \) are defined by \[ f(x) = x^2 - 3x + 4 \quad \text{and} \quad g(x) = 2x + 1, \quad \text{then the value of} \quad x \text{ for which} \quad f(x) = f \circ g(x) \text{ is} \]

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

Hint:

When solving functional equations, substitute the expression for the composite function and simplify step by step.

Answer 1

The Correct Option is B

Step 1: Equation for \( f \circ g(x) \).
We need to solve for \( x \) such that \( f(x) = f(g(x)) \). Substituting \( g(x) = 2x + 1 \) into the expression for \( f(x) \), we get: \[ f(g(x)) = (2x + 1)^2 - 3(2x + 1) + 4 \]
Step 2: Simplify and solve for \( x \).
Solving this equation, we find the values of \( x \) as \( -1 \) and \( \frac{2}{3} \), corresponding to option (B).