To find the initial value of the function [tex]\( f(x) = 9 \left( \frac{2}{3} \right)^x + 4 \)[/tex], we need to evaluate the function at [tex]\( x = 0 \)[/tex].
Step-by-step solution:
1. We start with the function given:
[tex]\[ f(x) = 9 \left( \frac{2}{3} \right)^x + 4 \][/tex]
2. Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = 9 \left( \frac{2}{3} \right)^0 + 4 \][/tex]
3. Simplify the term [tex]\( \left( \frac{2}{3} \right)^0 \)[/tex]:
[tex]\[ \left( \frac{2}{3} \right)^0 = 1 \][/tex]
4. Continue with the simplified expression:
[tex]\[ f(0) = 9 \cdot 1 + 4 \][/tex]
5. Perform the arithmetic:
[tex]\[ 9 \cdot 1 = 9 \][/tex]
[tex]\[ 9 + 4 = 13 \][/tex]
Thus, the initial value of the function [tex]\( f(x) = 9 \left( \frac{2}{3} \right)^x + 4 \)[/tex] is [tex]\(\boxed{13}\)[/tex].
