the domain and range of f(x)=(e^-x)-2
Start with the base function: y=e^x.
The range of y=e^x is (0, infinity).
If you have y=e^-x, that flips the function over the y axis, which does not change the range.
Now factor in the -2. That shifts the function down, so subtract 2 from the range: (-2, infinity)
Oops, forgot the domain.
e^x exists for all real values of x, so e^-x does also.
Therefore, the domain is (-infinity, infinity)
To find the domain and range of the function f(x) = (e^-x) - 2, let's start with the domain.
The domain of a function is the set of all possible input values, or x-values, for which the function is defined. In this case, since we have the exponential function e^(-x) as part of the function, we need to determine the domain of this exponential function.
For any exponential function of the form e^x, the domain is all real numbers, because the exponential function is defined for all real values of x. However, since we have a negative sign before the x term (e^-x), we need to consider any restrictions that it may introduce.
Unlike the natural logarithm function (ln(x)), which is not defined for x ≤ 0, the exponential function e^x is defined for all real numbers, including negative values. Therefore, there are no restrictions on the domain of f(x) = (e^-x) - 2, and the domain is all real numbers, (-∞, ∞).
Now, let's move on to finding the range of the function.
The range of a function is the set of all possible output values, or y-values, that the function can produce. To determine the range, we usually analyze the behavior of the function as the input values vary.
In this case, the function f(x) = (e^-x) - 2 is a composition of two functions: the exponential function e^-x and a constant shift of -2. The exponential function e^-x always produces positive values (since it's raising a positive base to a negative power), and then subtracting 2 from those positive values will still yield negative values.
So, the range of f(x) = (e^-x) - 2 is all real numbers less than -2, which can be represented as (-∞, -2).