To evaluate the function \( f(x) = \left( \frac{3}{4} \right)^{x+2} - 9 \) at \( x = 0 \), we can substitute \( x = 0 \) into the function:
\[ f(0) = \left( \frac{3}{4} \right)^{0+2} - 9 \]
This simplifies to:
\[ f(0) = \left( \frac{3}{4} \right)^{2} - 9 \]
Calculating \( \left( \frac{3}{4} \right)^{2} \):
\[ \left( \frac{3}{4} \right)^{2} = \frac{3^2}{4^2} = \frac{9}{16} \]
Now substituting that back into the expression for \( f(0) \):
\[ f(0) = \frac{9}{16} - 9 \]
To perform the subtraction, convert 9 to a fraction with a denominator of 16:
\[ 9 = \frac{144}{16} \]
Now, substitute this back in:
\[ f(0) = \frac{9}{16} - \frac{144}{16} = \frac{9 - 144}{16} = \frac{-135}{16} \]
Thus, the final result is:
\[ f(0) = -\frac{135}{16} \]
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