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

To determine which function is decreasing for all real values of $x$, we can analyze the derivatives of the given functions:

1. For $f(x) = e^x$, the derivative is $f'(x) = e^x$, which is always positive. Thus, this function is increasing.

2. For $f(x) = -e^{1-x}$, the derivative is: $f'(x) = -(-e^{1-x})(-1) = e^{1-x}$ This is also always positive, indicating the function is increasing.

3. For $f(x) = -e^{x-1}$, the derivative is: $f'(x) = -e^{x-1}$ This is always negative, indicating that this function is decreasing.

4. For $f(x) = -e^{-x}$, the derivative is: $f'(x) = -(-e^{-x})(-1) = e^{-x}$ This is also positive, indicating that this function is increasing.

Based on the analysis, the function that is decreasing for all real values of $x$ is:

$f(x) = -e^{x-1}$.