The function f is defined by f(y) = 2x + 1 Solve the equation f(x) = f^{-1}(x) Circle your answer - AQA - A-Level Maths Pure - Question 3 - 2022 - Paper 3

To solve the equation, first, we need to determine the inverse function f^{-1}(x). Given that f(y) = 2x + 1, we can express this as:

1. Set y = f(x):

   $y = 2x + 1$

2. Solve for x in terms of y:

   $y - 1 = 2x$

   $x = \frac{y - 1}{2}$

Thus, the inverse is represented as:

$f^{-1}(x) = \frac{x - 1}{2}$

Now, equate f(x) to f^{-1}(x):

$2x + 1 = \frac{x - 1}{2}$

3. Multiply both sides by 2 to eliminate the fraction:

   $2(2x + 1) = x - 1$

   Thus,

   $4x + 2 = x - 1$

4. Rearranging gives:

   $4x - x = -1 - 2$

   $3x = -3$

   $x = -1$

5. Therefore, the solution to the equation is:

   $x = -1$

This indicates the correct answer is -1.