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Given that \( f(x) = x^2 - 10 \) and \( g(x) = 9x + 1 \), find \( (fg)\left(-\frac{1}{9}\right) \), if it exists.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. \( (fg)\left(-\frac{1}{9}\right) = \square \) (Simplify your answer.)
B. The value for [tex]\( (fg)\left(-\frac{1}{9}\right) \)[/tex] does not exist.



Answer :

GinnyAnswer
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]

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