A student is searching for a solution to the equation $f(x) = 0$ - AQA - A-Level Maths Pure - Question 2 - 2020 - Paper 1

To determine which function does not have a root between -1 and 1, we analyze:

1. For $f(x) = \frac{1}{x}$: This function does not exist at $x = 0$, and it does change sign between -1 and 1. So this is the function where the conclusion is incorrect.

2. For $f(x) = x$: The function is linear with a root at $x = 0$, so the conclusion holds.

3. For $f(x) = x^3$: This is a cubic function that crosses the x-axis at $x = 0$, affirming the conclusion.

4. For $f(x) = \frac{2x + 1}{x + 2}$: Evaluating this at -1 and 1 shows it changes sign.

Thus, the answer is: $f(x) = \frac{1}{x}$.