To find \((fg)\left(-\frac{1}{9}\right)\), we will follow these steps systematically:
1. Determine the function \(g(x)\)
- Given \( g(x) = 9x + 1 \)
- We need to substitute \(x = -\frac{1}{9}\) into \( g(x) \).
2. Evaluate \( g\left( -\frac{1}{9} \right) \)
- Substitute \( x = -\frac{1}{9} \) into \( g(x) = 9x + 1 \):
[tex]\[
g\left( -\frac{1}{9} \right) = 9 \left( -\frac{1}{9} \right) + 1
\][/tex]
[tex]\[
g\left( -\frac{1}{9} \right) = -1 + 1 = 0
\][/tex]
- So, \( g\left( -\frac{1}{9} \right) = 0 \).
3. Determine the function \( f(x) \)
- Given \( f(x) = x^2 - 10 \)
- We need to substitute \( x = 0 \) (the result from the previous step) into \( f(x) \).
4. Evaluate \( f(g(x)) \) or \( f(g\left( -\frac{1}{9} \right)) \)
- We need to find \( f(0) \):
[tex]\[
f(0) = 0^2 - 10 = 0 - 10 = -10
\][/tex]
So, \((fg)\left( -\frac{1}{9} \right) = -10\).
Thus, the correct choice is:
A. [tex]\((fg)\left( -\frac{1}{9} \right) = -10\)[/tex]
