To determine the range of the function [tex]\( f(x) = 2 \cdot 3^x \)[/tex], let's analyze its behavior.
1. Understanding the Function:
- [tex]\( f(x) = 2 \cdot 3^x \)[/tex] is an exponential function where the base of the exponent, 3, is greater than 1.
- The coefficient 2 is a positive constant that scales the exponential function.
2. Behavior of Exponential Functions:
- For [tex]\( 3^x \)[/tex], where [tex]\( 3 \)[/tex] is the base, as [tex]\( x \)[/tex] increases, [tex]\( 3^x \)[/tex] grows rapidly.
- As [tex]\( x \)[/tex] decreases (negative values), [tex]\( 3^x \)[/tex] approaches 0 but never becomes 0. It gets very close to 0.
3. Impact of the Coefficient 2:
- The overall function [tex]\( f(x) = 2 \cdot 3^x \)[/tex] scales the values of [tex]\( 3^x \)[/tex] by a factor of 2.
- Therefore, as [tex]\( x \)[/tex] increases, [tex]\( 2 \cdot 3^x \)[/tex] grows very rapidly.
- As [tex]\( x \)[/tex] decreases, [tex]\( 2 \cdot 3^x \)[/tex] gets close to 0 but always remains positive (since 2 times a positive number is still positive).
4. Evaluating the Range:
- For very large negative [tex]\( x \)[/tex]: [tex]\( f(x) = 2 \cdot 3^x \)[/tex] approaches 2 times a very small positive number (close to zero).
- For very large positive [tex]\( x \)[/tex]: [tex]\( f(x) = 2 \cdot 3^x \)[/tex] approaches very large positive values.
Thus, the function [tex]\( f(x) = 2 \cdot 3^x \)[/tex] always results in a positive output and never touches 0. Therefore, the best description of the range of [tex]\( f(x) = 2 \cdot 3^x \)[/tex] is:
[tex]\[ y > 0 \][/tex]
