Question

"Determine the domain on which the following function is decreasing. Also, how would you write it?"

a) The domain on which the function is decreasing is (-[infinity], 0), and you would write it as f(x) = e^(-x).
b) The domain on which the function is decreasing is (-[infinity], [infinity]), and you would write it as f(x) = 1 - x.
c) The domain on which the function is decreasing is (0, [infinity]), and you would write it as f(x) = -ln(x).
d) The domain on which the function is decreasing is (0, [infinity]), and you would write it as f(x) = e^(1/x).

Answer 1

To determine the domain on which a function is decreasing, we analyze the slope of the function. The correct answer is option c), where the domain is (0, [infinity]) and the function is written as f(x) = -ln(x).

Step-by-step explanation:

a) The function f(x) = e^(-x) is a decreasing function on the domain (-[infinity], 0). The negative sign in front of x in the exponent indicates that the function is decreasing as x increases.

b) The function f(x) = x^2 is an increasing function for x > 0, and it is decreasing for x < 0. Therefore, it doesn't have a specific domain where it is decreasing.

c) The function f(x) = -ln(x) is a decreasing function on the domain (0, [infinity]). The logarithm function has a negative slope, which means it is decreasing.

d) The function f(x) = e^(1/x) is not defined at x = 0, and it is decreasing for x > 0. Therefore, the domain on which the function is decreasing is (0, [infinity]).

Based on this analysis, option c) is the correct answer. The domain on which the function is decreasing is (0, [infinity]), and it can be written as f(x) = -ln(x).