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 need to examine the derivatives of each function:

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

2. For $f(x) = -e^{1-x}$, we can calculate the derivative: $f'(x) = -(-e^{1-x})' = e^{1-x}$ The derivative is positive, indicating this function is also increasing.

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

4. For $f(x) = -e^{-x}$, the derivative is: $f'(x) = -(-e^{-x})' = e^{-x}$ The derivative is positive, showing this function is increasing.

From the above analysis, we conclude that the only function that is decreasing for all real values of $x$ is $f(x) = -e^{x-1}$.