To find \((f \cdot g)(x)\), we can multiply the two functions \(f(x)\) and \(g(x)\) together.
Given:
\[ f(x) = \sqrt{x + 3} - 2 \] \[ g(x) = \sqrt{x + 3} \]
Now, multiplying \(f(x)\) and \(g(x)\):
\[ (f \cdot g)(x) = f(x) \cdot g(x) = (\sqrt{x + 3} - 2) \cdot \sqrt{x + 3} \]
Using the distributive property (FOIL method):
\[ (f \cdot g)(x) = \sqrt{x + 3} \cdot \sqrt{x + 3} - 2 \cdot \sqrt{x + 3} \]
Now calculate each term:
-
The first term: \[ \sqrt{x + 3} \cdot \sqrt{x + 3} = x + 3 \]
-
The second term: \[ -2 \cdot \sqrt{x + 3} = -2\sqrt{x + 3} \]
Putting it all together:
\[ (f \cdot g)(x) = (x + 3) - 2\sqrt{x + 3} \]
So, we can express this as:
\[ (f \cdot g)(x) = x + 3 - 2\sqrt{x + 3} \]
To put it in the required format \(x + ? + (?)\sqrt{x + 3}\):
- The first missing value is \(3\).
- The second missing value is \(-2\).
Thus,
\[ (f \cdot g)(x) = x + 3 - 2\sqrt{x + 3} \]
The final answer is:
\[ (f \cdot g)(x) = x + 3 + (-2)\sqrt{x + 3} \]
The missing values are \(3\) and \(-2\).