To find the domain and range of the exponential function f(x) = 2^(1x) (assuming 1x to simply be x as the multiplication by 1 does not change the value of x), we will analyze the behavior of the function.
**Domain:**
The domain of a function is the set of all possible inputs (x-values) that the function can accept. For the exponential function f(x) = 2^x, the input x can be any real number. There's no restriction on the value of x because you can raise 2 to the power of any real number, whether it is positive, negative, or zero. Therefore, the domain of f(x) = 2^x is all real numbers.
Correct domain: (All real numbers)
**Range:**
The range of a function is the set of all possible outputs (y-values) that the function can produce. With f(x) = 2^x, for any real number x, 2 raised to the power of x will always be positive because any non-zero number raised to any power is positive; moreover, 2^x approaches 0 as x approaches negative infinity but never actually reaches 0. It also increases without bound as x increases. Therefore, the range of f(x) = 2^x is all positive real numbers, including values very close to zero, but not including zero itself.
Correct range: (0 < y < ∞)
There seems to be an error in the range options you provided. None of the options accurately represents the range of the function. However, if the last option intended to represent positive real numbers (with a possible typo), the correct range would be ...(0 < y < ∞)... with the understanding that it should actually be “(0 < y < ∞)” for mathematical correctness.
Based on this analysis, the correct choice that identifies both the domain and range would be:
Domain: (All real numbers)
Range: (0 < y < ∞) (this is assuming a typo in the options provided and using the corrected range)
