Which one of these functions is decreasing for all real values of x? Circle your answer - AQA - A-Level Maths Mechanics - Question 1 - 2020 - Paper 2

To determine which function is decreasing for all real values of $x$, we need to analyze each function and its derivative.

1. $f(x) = e^x$ increases for all $x$ since its derivative $f'(x) = e^x > 0$.
2. $f(x) = -e^{1-x}$: The derivative is $f'(x) = e^{1-x} > 0$, hence the function is decreasing.
3. $f(x) = -e^{x-1}$: The derivative is $f'(x) = -e^{x-1} < 0$, indicating the function is increasing.
4. $f(x) = -e^{-x}$: The derivative $f'(x) = e^{-x}$ is always positive, indicating the function is increasing.

Thus, the correct answer is: $f(x) = -e^{1-x}$, which is decreasing for all real values of $x$.