To determine the domain and range of the function [tex]\( f(x) = 2(3)^x + 3 \)[/tex], we can analyze the components separately and understand how they interact within the function.
### Domain
1. The exponential function [tex]\( 3^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. It takes any real number and maps it to a positive real number.
2. When we multiply [tex]\( 3^x \)[/tex] by 2, we obtain [tex]\( 2(3)^x \)[/tex], which is also defined for all real numbers [tex]\( x \)[/tex] because it just scales the output of [tex]\( 3^x \)[/tex] by 2.
3. Finally, adding 3 to [tex]\( 2(3)^x \)[/tex], we get [tex]\( 2(3)^x + 3 \)[/tex]. The addition of 3 is also defined for all real numbers [tex]\( x \)[/tex].
Combining these facts, there are no restrictions on the values [tex]\( x \)[/tex] can take. Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
In interval notation, the domain is:
[tex]\[ (-\infty, \infty) \][/tex]
### Range
1. Let's consider the range of the exponential component [tex]\( 2(3)^x \)[/tex]. Exponential functions like [tex]\( 3^x \)[/tex] are always positive and grow rapidly as [tex]\( x \)[/tex] increases. Specifically, [tex]\( 3^x \)[/tex] approaches 0 as [tex]\( x \)[/tex] approaches negative infinity, and it grows towards positive infinity as [tex]\( x \)[/tex] increases.
2. By multiplying [tex]\( 3^x \)[/tex] by 2, [tex]\( 2(3)^x \)[/tex] will still be positive and will also grow towards infinity as [tex]\( x \)[/tex] increases. The minimum value of [tex]\( 2(3)^x \)[/tex] occurs as [tex]\( x \)[/tex] approaches negative infinity, still approaching 0.
3. Adding 3 to [tex]\( 2(3)^x \)[/tex] shifts the entire curve upwards by 3 units. Therefore, the output, [tex]\( f(x) = 2(3)^x + 3 \)[/tex], will always be greater than 3. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 2(3)^x + 3 \)[/tex] approaches 3, but never actually reaches it. As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] grows towards positive infinity.
Thus, the range of [tex]\( f(x) \)[/tex] starts just above 3 and extends to positive infinity.
In interval notation, the range is:
[tex]\[ (3, \infty) \][/tex]
To summarize:
- The domain of [tex]\( f(x) = 2(3)^x + 3 \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
- The range of [tex]\( f(x) = 2(3)^x + 3 \)[/tex] is [tex]\( (3, \infty) \)[/tex].
