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 satisfy the Intermediate Value Theorem, we evaluate each option:

1. $f(x) = \frac{1}{x}$: This function is undefined at $x = 0$, thus there cannot be a root in the interval $(-1, 1)$ as $f(-1) = -1$ and $f(1) = 1$, yet it does not cross the x-axis.

2. $f(x) = x$: This function does have a root at $x = 0$, hence the conclusion is correct for this function.

3. $f(x) = x^3$: This function has a root at $x = 0$, hence the conclusion is correct for this function as well.

4. $f(x) = \frac{2x + 1}{x + 2}$: Evaluating at $f(-1)$ gives $0$ and $f(1) = \frac{3}{3} = 1$, hence there is a change of sign.

Thus, the function for which the conclusion is incorrect is $f(x) = \frac{1}{x}$.