Question

If \( f(x) = x^2 - 3x + 4 \) and \( f(x) = f(2x + 1) \), then \( x = \)

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

Hint:

For problems involving equations of functions, equate the two expressions and solve for the unknown variable as you would in a standard algebraic equation.

Answer 1

The Correct Option is A

Step 1: Equating the two functions.
We are given \( f(x) = f(2x + 1) \), so \[ x^2 - 3x + 4 = (2x + 1)^2 - 3(2x + 1) + 4 \] Step 2: Simplify and solve for \( x \).
After expanding and simplifying the equation, we get: \[ x^2 - 3x + 4 = 4x^2 + 4x + 1 - 6x - 3 + 4 \] Simplifying further gives: \[ x^2 - 3x + 4 = 4x^2 - 2x + 2 \] Solving this quadratic equation results in \( x = -1 \) and \( x = \frac{2}{3} \).
Step 3: Conclusion.
The correct answer is (A) \( -1, \frac{2}{3} \).