1. The problem asks to identify an irrational number from the given options: $\sqrt{11}$, $4.2$, $\frac{3}{4}$, and $\sqrt{81}$.\n\n2. An irrational number is a number that cannot be expressed as a ratio of two integers, meaning it cannot be written as a simple fraction. Its decimal form is non-terminating and non-repeating.\n\n3. Let's analyze each option:\n- $\sqrt{11}$: Since 11 is not a perfect square, $\sqrt{11}$ is an irrational number.\n- $4.2$: This is a terminating decimal and can be written as $\frac{42}{10}$, so it is rational.\n- $\frac{3}{4}$: This is a fraction of two integers, so it is rational.\n- $\sqrt{81}$: Since 81 is a perfect square ($9^2$), $\sqrt{81} = 9$, which is rational.\n\n4. Therefore, the irrational number among the options is $\sqrt{11}$.