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

To determine which function does not have an inverse, we need to consider the nature of each function:

1. $f(x) = x^3$: This function is a cubic polynomial. It is one-to-one because it is strictly increasing for all $x \in \mathbb{R}$. Therefore, it has an inverse.

2. $f(x) = 2x + 1$: This is a linear function with a slope of 2, which is non-zero. Linear functions are also one-to-one, thus it has an inverse.

3. $f(x) = x^2$: This function is a quadratic polynomial. It is not one-to-one since it fails the horizontal line test (e.g., $f(1) = 1$ and $f(-1) = 1$). Therefore, it does not have an inverse.

4. $f(x) = e^x$: The exponential function is strictly increasing and one-to-one for all $x \in \mathbb{R}$, hence it has an inverse.

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