Each of these functions has domain $x \in \mathbb{R}$ Which function does not have an inverse? Circle your answer - AQA - A-Level Maths Mechanics - Question 3 - 2019 - Paper 2

To determine which of the given functions does not have an inverse, we need to analyze the behavior of each function:

1. $f(x) = x^3$: This function is one-to-one, meaning every output corresponds to only one input. Therefore, it has an inverse.

2. $f(x) = 2x + 1$: This is a linear function with a non-zero slope. It is also one-to-one and has an inverse.

3. $f(x) = x^2$: This function is not one-to-one since it gives the same output for both positive and negative inputs (e.g., $f(2) = 4$ and $f(-2) = 4$). Therefore, this function does not have an inverse.

4. $f(x) = e^x$: This exponential function is one-to-one and has an inverse (the natural logarithm).

Based on the analysis, the function that does not have an inverse is $f(x) = x^2$.