Certainly! Let's analyze the function [tex]\( y = 4 \cdot 3^x \)[/tex].
### Step-by-Step Solution:
1. Understanding the Function:
The function [tex]\( y = 4 \cdot 3^x \)[/tex] is an exponential function with the base [tex]\( 3 \)[/tex] and a coefficient of [tex]\( 4 \)[/tex].
2. Domain:
The domain of a function refers to all possible input values (values of [tex]\( x \)[/tex]) for which the function is defined. For exponential functions such as [tex]\( y = 4 \cdot 3^x \)[/tex], [tex]\( x \)[/tex] can be any real number. There are no restrictions on [tex]\( x \)[/tex]. Therefore, the domain is:
[tex]\[ \text{Domain: All Real Numbers} \][/tex]
3. Range:
The range of a function refers to all possible output values (values of [tex]\( y \)[/tex]) that the function can produce. For the exponential function [tex]\( y = 4 \cdot 3^x \)[/tex]:
- The base [tex]\( 3 \)[/tex] is a positive number greater than 1.
- An exponential function of the form [tex]\( a \cdot b^x \)[/tex], where [tex]\( a > 0 \)[/tex] and [tex]\( b > 1 \)[/tex], will always yield positive values for [tex]\( y \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( 3^x \to 0 \)[/tex], but [tex]\( y \)[/tex] never actually reaches 0; it gets arbitrarily close to it from the positive side.
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \to \infty \)[/tex].
Therefore, the output [tex]\( y \)[/tex] is always greater than 0. Hence, the range is:
[tex]\[ \text{Range: } y > 0 \][/tex]
### Conclusion:
Based on our step-by-step analysis, we can determine that the correct domain and range for the function [tex]\( y = 4 \cdot 3^x \)[/tex] are:
- Domain: All Real Numbers
- Range: [tex]\( y > 0 \)[/tex]
So the correct answer is:
[tex]\[ \text{Domain: All Real Numbers} \][/tex]
[tex]\[ \text{Range: } y > 0 \][/tex]
