To determine the range of the function [tex]\( f(x) = 3^x + 9 \)[/tex], we need to analyze the behavior of this function for different values of [tex]\( x \)[/tex].
1. Understand the base function [tex]\( 3^x \)[/tex]:
- For [tex]\( x = 0 \)[/tex], [tex]\( 3^0 = 1 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( 3^x \)[/tex] increases exponentially.
- As [tex]\( x \)[/tex] decreases (i.e., for negative [tex]\( x \)[/tex]), [tex]\( 3^x \)[/tex] approaches [tex]\( 0 \)[/tex] but remains positive because an exponential function with an exponent going to negative infinity approaches zero.
2. Analyse the transformation:
- The function [tex]\( f(x) = 3^x + 9 \)[/tex] vertically shifts the base function [tex]\( 3^x \)[/tex] by 9 units upward.
- Consequently, the minimum value of [tex]\( f(x) \)[/tex] occurs when [tex]\( 3^x \)[/tex] is at its minimum, which is when [tex]\( 3^x = 1 \)[/tex] (i.e., [tex]\( x = 0 \)[/tex]).
- Therefore, the smallest value of [tex]\( f(x) \)[/tex] is [tex]\( 1 + 9 = 10 \)[/tex].
3. Range determination:
- Given that [tex]\( 3^x \geq 0 \)[/tex] for all [tex]\( x \)[/tex], [tex]\( f(x) = 3^x + 9 \)[/tex] will always be greater than 9.
- Hence, [tex]\( f(x) \)[/tex] will never equal or drop below 9.
Thus, the range of the function [tex]\( f(x) = 3^x + 9 \)[/tex] is [tex]\( \{y \mid y > 9\} \)[/tex].
The correct choice is: [tex]\(\{y \mid y > 9\}\)[/tex].
