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B) The exponential function [tex]f(x) = 2^x[/tex]

Domain: [tex]$\qquad$[/tex] All real numbers [tex]$\mathbb{R}$[/tex]

Range: [tex]$\qquad$[/tex] All positive real numbers [tex]$(0, \infty)$[/tex]

Sagot :

GinnyAnsweridn
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]
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