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 values of $x$, we can analyze the derivatives of each function.

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

2. For the second function, $f(x) = -e^{1-x}$, the derivative is: f'(x) = -(-e^{1-x}) rac{d}{dx}(1-x) = e^{1-x}
   This derivative is positive, indicating that this function is increasing.

3. For the third function, $f(x) = -e^{-x}$, the derivative is: $f'(x) = -(-e^{-x}) = e^{-x}$
   This function's derivative is also positive, so it is increasing as well.

4. For the fourth function, which is $f(x) = -e^{-x}$, the analysis remains the same: $f'(x) = -(-e^{-x}) = e^{-x}$
   As with the third function, it is increasing.

Upon examining the second function, $f(x) = -e^{1-x}$, we realize that for $x o ext{infinity}$, it approaches $- ext{infinity}$, which maintains a negative slope and thus exhibits decreasing behavior overall. Hence, the correct answer is $f(x) = -e^{1-x}$.