To determine the domain and range of the exponential function [tex]\( f(x) = 2^x \)[/tex], let's analyze the function step by step.
### Domain:
The domain of a function consists of all possible input values (x-values) for which the function is defined.
1. Consider the function [tex]\( f(x) = 2^x \)[/tex].
2. An exponential function is defined for all real numbers because you can raise 2 to any real exponent without restriction.
3. Therefore, the domain of [tex]\( f(x) = 2^x \)[/tex] is all real numbers.
So, the domain is:
[tex]\[ \text{Domain: } (-\infty, \infty) \][/tex]
### Range:
The range of a function consists of all possible output values (y-values) as [tex]\( x \)[/tex] varies over the domain.
1. Consider the function [tex]\( f(x) = 2^x \)[/tex].
2. By definition, [tex]\( 2^x \)[/tex] is always positive for any real number [tex]\( x \)[/tex].
3. It approaches 0 asymptotically as [tex]\( x \)[/tex] approaches negative infinity, but never actually reaches 0.
4. As [tex]\( x \)[/tex] increases, [tex]\( 2^x \)[/tex] grows exponentially without bound.
So, the output will never be zero or negative, and it can take on any positive value.
Therefore, the range is:
[tex]\[ \text{Range: } (0, \infty) \][/tex]
### Summary:
- Domain: [tex]\( (-\infty, \infty) \)[/tex]
- Range: [tex]\( (0, \infty) \)[/tex]
