Let's analyze the exponential function [tex]\( y = 3 \cdot 5^x \)[/tex] to determine its domain and range.
### Domain:
The domain of a function refers to the set of all possible input values (x-values) that the function can accept. For an exponential function of the form [tex]\( y = a \cdot b^x \)[/tex] where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, the value [tex]\( b^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
Since there are no restrictions on [tex]\( x \)[/tex] (it can be any real number), the domain of the function [tex]\( y = 3 \cdot 5^x \)[/tex] is:
[tex]\[
\text{all real numbers}
\][/tex]
### Range:
The range of a function refers to the set of all possible output values (y-values) that the function can produce. For an exponential function [tex]\( y = a \cdot b^x \)[/tex]:
1. Exponent [tex]\( x \)[/tex]: For any real number [tex]\( x \)[/tex], [tex]\( 5^x \)[/tex] is always positive.
2. Constant [tex]\( a = 3 \)[/tex]: Since 3 is a positive constant and multiplying a positive number [tex]\( 5^x \)[/tex] by a positive constant results in another positive number, [tex]\( y = 3 \cdot 5^x \)[/tex] will always be positive.
Therefore, the function [tex]\( y = 3 \cdot 5^x \)[/tex] can produce any positive real number as an output.
The range of the function [tex]\( y = 3 \cdot 5^x \)[/tex] is:
[tex]\[
\text{all positive real numbers}
\][/tex]
### Final Answer:
The domain of the function [tex]\( y = 3 \cdot 5^x \)[/tex] is [tex]\(\boxed{\text{all real numbers}}\)[/tex].
The range of the function [tex]\( y = 3 \cdot 5^x \)[/tex] is [tex]\(\boxed{\text{all positive real numbers}}\)[/tex].
DONE
