Find the domain, range and horizontal asymptote
f(x)=2^-x-2
domain: all reals
range all reals > 0
h.a.: y=0
To find the domain, range, and horizontal asymptote of the function f(x) = 2^(-x) - 2, we need to analyze its properties.
1. Domain:
The domain refers to the set of all possible input values for the function. In this case, since we are dealing with exponentiation, the only restriction comes from the base of the exponent, which is 2. The base 2^x is defined for all real numbers x, so there are no restrictions on the domain. Therefore, the domain of the function f(x) is (-∞, ∞), meaning it is defined for all real numbers.
2. Range:
The range refers to the set of all possible output values for the function. To determine the range, we need to consider the behavior of the function as x approaches positive and negative infinity.
As x approaches positive infinity, 2^(-x) approaches zero, and subtracting 2 from zero gives us -2. Therefore, as x approaches positive infinity, f(x) approaches -2.
As x approaches negative infinity, 2^(-x) approaches positive infinity, and subtracting 2 from positive infinity gives us positive infinity. Therefore, as x approaches negative infinity, f(x) approaches positive infinity.
Based on these observations, we can say that the range of the function f(x) is (-∞, -2].
3. Horizontal Asymptote:
A horizontal asymptote is a horizontal line that a function approaches as x approaches positive or negative infinity. To determine the horizontal asymptote for the given function, we look at the behavior of the function as x approaches positive and negative infinity.
As x approaches positive infinity, f(x) approaches -2, which means y = -2 is the horizontal asymptote.
As x approaches negative infinity, f(x) approaches positive infinity, which means there is no horizontal asymptote in the negative direction.
Therefore, the horizontal asymptote for the function f(x) is y = -2.
Summary:
- The domain of the function f(x) = 2^(-x) - 2 is (-∞, ∞).
- The range of the function f(x) is (-∞, -2].
- The horizontal asymptote of the function f(x) is y = -2.