To determine the range of the function [tex]\( f(x) = 3^x + 9 \)[/tex], let's analyze its behavior step-by-step.
1. Understand the Function:
- The function [tex]\( f(x) = 3^x + 9 \)[/tex] consists of two parts: the exponential part [tex]\( 3^x \)[/tex] and the constant part [tex]\( 9 \)[/tex].
2. Behavior of the Exponential Function [tex]\( 3^x \)[/tex]:
- As [tex]\( x \)[/tex] increases, [tex]\( 3^x \)[/tex] increases exponentially. For very large values of [tex]\( x \)[/tex], [tex]\( 3^x \)[/tex] becomes very large.
- As [tex]\( x \)[/tex] decreases and approaches negative infinity, [tex]\( 3^x \)[/tex] approaches [tex]\( 0 \)[/tex].
3. Combining Both Parts:
- Since [tex]\( 3^x \)[/tex] approaches [tex]\( 0 \)[/tex] as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) = 3^x + 9 \)[/tex] approaches [tex]\( 9 \)[/tex].
- Because the exponential function [tex]\( 3^x \)[/tex] is always positive, [tex]\( 3^x > 0 \)[/tex] for all real [tex]\( x \)[/tex]. Thus, [tex]\( 3^x + 9 > 9 \)[/tex].
4. Conclusion on the Range:
- The minimum value that [tex]\( f(x) \)[/tex] can approach is [tex]\( 9 \)[/tex], but it never actually reaches [tex]\( 9 \)[/tex]; it only gets arbitrarily close to [tex]\( 9 \)[/tex] as [tex]\( x \)[/tex] approaches negative infinity.
- For any other real number [tex]\( x \)[/tex], [tex]\( f(x) \)[/tex] will be greater than [tex]\( 9 \)[/tex].
So, the range of [tex]\( f(x) = 3^x + 9 \)[/tex] is all real numbers greater than [tex]\( 9 \)[/tex].
Thus, the correct choice is:
[tex]\[ \{ y \mid y > 9 \} \][/tex]
