- Identify that \(x=6\) falls within the interval \(-8 \leq x \leq 8\).
- Select the appropriate expression for this interval: \(f(x) = 6x^2 + 10\).
- Substitute \(x=6\) into the expression to compute: \(6(6)^2 + 10 = 216 + 10 = 226\).
- The final result is \(\boxed{226}\) (Option A).
Explanation
- Determining the Correct Function Interval
To evaluate \(f(6)\) for the given piecewise function, we first identify the interval to which \(x=6\) belongs. The function \(f(x)\) is defined as follows:
\[ f(x) = \begin{cases} -3x^2 - 4 & x < -8 \ 6x^2 + 10 & -8 \leq x \leq 8 \ 9 & x > 8 \end{cases} \]
Comparing \(x=6\) to the given intervals, we see that \(-8 \leq 6 \leq 8\). Therefore, we must use the second part of the function definition, which is \(f(x) = 6x^2 + 10\).
- Calculating the Function Value
Now we substitute \(x=6\) into the expression \(f(x) = 6x^2 + 10\):
\[ f(6) = 6(6)^2 + 10 \]
First, calculate the square of 6:
\[ 6^2 = 36 \]
Next, multiply this result by 6:
\[ 6 \times 36 = 216 \]
Finally, add 10 to the result:
\[ 216 + 10 = 226 \]
Thus, \(f(6) = 226\). Comparing this to the given choices, it corresponds to option A.
Examples
Piecewise functions are frequently used to model real-world scenarios where different rules apply under different conditions. For example, a shipping company might charge different rates based on the weight of a package: a flat rate for lightweight items, a weight-based rate for medium items, and a different flat rate for very heavy items. Evaluating such a function at a specific weight is identical to the process we used here to find the value of \(f(6)\), ensuring that the correct rule is applied to the specific input value.