To identify the domain and range of the function [tex]\( y = 3 \cdot 5^x \)[/tex], let's break the problem down step-by-step.
### Domain:
The domain of a function consists of all the input values (x-values) for which the function is defined. In this case, the function is [tex]\( y = 3 \cdot 5^x \)[/tex].
1. The function involves the term [tex]\( 5^x \)[/tex].
2. Exponential functions of the form [tex]\( a^x \)[/tex] (where [tex]\( a > 0 \)[/tex]) are defined for all real values of [tex]\( x \)[/tex].
Therefore, for the function [tex]\( y = 3 \cdot 5^x \)[/tex], [tex]\( 5^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
- Domain: All real numbers.
### Range:
The range of a function consists of all the possible output values (y-values) of the function.
1. Consider the base term [tex]\( 5^x \)[/tex]:
- Exponential functions [tex]\( 5^x \)[/tex] only produce positive values. That is, [tex]\( 5^x > 0 \)[/tex] for all real [tex]\( x \)[/tex].
2. The function [tex]\( y = 3 \cdot 5^x \)[/tex]:
- Since [tex]\( 5^x > 0 \)[/tex], multiplying by 3 will still give positive values.
- Therefore, [tex]\( y = 3 \cdot 5^x \)[/tex] is always positive.
Hence, the function [tex]\( y = 3 \cdot 5^x \)[/tex] will produce all positive real numbers.
- Range: All positive real numbers.
In conclusion:
- The domain of the function [tex]\( y = 3 \cdot 5^x \)[/tex] is all real numbers.
- The range of the function [tex]\( y = 3 \cdot 5^x \)[/tex] is all positive real numbers.
