- Calculate the inner function value: \(g(-2) = (-2)^2 + 2 = 6\).
- Substitute the inner result into the outer function: \(f(6) = -\frac{1}{2} (6)^2 + 5(6)\).
- Evaluate the arithmetic: \(-18 + 30 = 12\).
- The final result is \(\boxed{12}\).
Explanation
- Problem Analysis and Strategy
The problem asks to evaluate the composite function \(f(g(-2))\) given the functions \(f(x) = -\frac{1}{2} x^2 + 5x\) and \(g(x) = x^2 + 2\). A composite function \(f(g(x))\) is evaluated by first computing the value of the inner function, \(g(x)\), at the given input, and then using that result as the input for the outer function, \(f(x)\). Here, we must evaluate \(g(-2)\) first, and then plug that result into \(f(x)\).
- Evaluating the Inner Function \(g(-2)\)
First, we evaluate the inner function \(g(x) = x^2 + 2\) at \(x = -2\).
Substituting \(-2\) for \(x\):
\[g(-2) = (-2)^2 + 2 = 4 + 2 = 6.\]
So, \(g(-2) = 6\).
- Evaluating the Outer Function \(f(6)\) and Conclusion
Now, we take the result from the previous step, which is \(6\), and use it as the input for the outer function \(f(x) = -\frac{1}{2} x^2 + 5x\).
Substituting \(6\) for \(x\):
\[f(6) = -\frac{1}{2} (6)^2 + 5(6).\]
Evaluating the exponent:
\[f(6) = -\frac{1}{2} (36) + 5(6).\]
Performing the multiplications:
\[f(6) = -18 + 30 = 12.\]
Therefore, the value of the composite function \(f(g(-2))\) is \(12\). Comparing this to the given options (\(-28\), \(-12\), \(12\), \(146\)), the correct value is \(12\).
Examples
Composite functions are frequently used in modeling real-world processes. For example, if the cost of producing a quantity of goods is a function of the number of labor hours, and the number of labor hours is a function of the units produced, you would use a composite function to determine the total cost directly from the units produced. Understanding how to evaluate these functions allows businesses to optimize production efficiency and predict costs accurately.