Answer :
To determine the domain and range of the exponential function [tex]\( y = 1.5(2)^x \)[/tex], let's start by understanding the properties of this type of function step by step:
1. Identify the Function Type:
The function [tex]\( y = 1.5(2)^x \)[/tex] is an exponential function. Exponential functions typically have the form [tex]\( y = a(b)^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.
2. Determine the Domain:
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. Exponential functions are defined for all real numbers because you can raise any positive number to any real number power. Therefore, the domain of [tex]\( y = 1.5(2)^x \)[/tex] is:
[tex]\[ \text{Domain: All Real Numbers} \][/tex]
3. Determine the Range:
The range of a function refers to the set of all possible output values (y-values). To determine this, consider what happens to [tex]\( y = 1.5(2)^x \)[/tex] as [tex]\( x \)[/tex] varies:
- When [tex]\( x \)[/tex] is very large (positive), [tex]\( 2^x \)[/tex] becomes very large, and thus [tex]\( y \)[/tex] becomes very large.
- When [tex]\( x \)[/tex] is very large (negative), [tex]\( 2^x \)[/tex] approaches 0, but it never reaches 0 or becomes negative, meaning [tex]\( y \)[/tex] approaches 1.5 times 0, still keeping [tex]\( y \)[/tex] positive.
Hence, the output [tex]\( y \)[/tex] will always be greater than 0 because [tex]\( 1.5(2)^x \)[/tex] never touches or goes below zero. Thus, the range of the function is:
[tex]\[ \text{Range: } y > 0 \][/tex]
Given these observations, the correct choices for the domain and range of the function [tex]\( y = 1.5(2)^x \)[/tex] are:
[tex]\[ \text{Domain: All Real Numbers} \hspace{1cm} \text{Range: } y > 0 \][/tex]
1. Identify the Function Type:
The function [tex]\( y = 1.5(2)^x \)[/tex] is an exponential function. Exponential functions typically have the form [tex]\( y = a(b)^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.
2. Determine the Domain:
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. Exponential functions are defined for all real numbers because you can raise any positive number to any real number power. Therefore, the domain of [tex]\( y = 1.5(2)^x \)[/tex] is:
[tex]\[ \text{Domain: All Real Numbers} \][/tex]
3. Determine the Range:
The range of a function refers to the set of all possible output values (y-values). To determine this, consider what happens to [tex]\( y = 1.5(2)^x \)[/tex] as [tex]\( x \)[/tex] varies:
- When [tex]\( x \)[/tex] is very large (positive), [tex]\( 2^x \)[/tex] becomes very large, and thus [tex]\( y \)[/tex] becomes very large.
- When [tex]\( x \)[/tex] is very large (negative), [tex]\( 2^x \)[/tex] approaches 0, but it never reaches 0 or becomes negative, meaning [tex]\( y \)[/tex] approaches 1.5 times 0, still keeping [tex]\( y \)[/tex] positive.
Hence, the output [tex]\( y \)[/tex] will always be greater than 0 because [tex]\( 1.5(2)^x \)[/tex] never touches or goes below zero. Thus, the range of the function is:
[tex]\[ \text{Range: } y > 0 \][/tex]
Given these observations, the correct choices for the domain and range of the function [tex]\( y = 1.5(2)^x \)[/tex] are:
[tex]\[ \text{Domain: All Real Numbers} \hspace{1cm} \text{Range: } y > 0 \][/tex]
