To find \( f(12.5) \), we need to determine which part of the piecewise function \( f(x) \) applies when \( x = 12.5 \).

The function \( f(x) \) is defined as:

\[
f(x) = 
\begin{cases}
9 - x^2, & \text{if } x < 2 \\
\sqrt{x} + 7, & \text{if } 2 \leq x < 10 \\
|x - 4|, & \text{if } x \geq 10 
\end{cases}
\]

Since \( x = 12.5 \) falls in the interval \( x \geq 10 \), we use the third piece of the function:

\[
f(x) = |x - 4|
\]

Now, plug in \( x = 12.5 \):

\[
f(12.5) = |12.5 - 4|
\]

\[
f(12.5) = |8.5| = 8.5
\]

Therefore, \( f(12.5) = 8.5 \).

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

1. What is \( f(0) \) for the given function?
2. How does the value of \( f(x) \) change for \( x \) values greater than 10?
3. What is the range of the function \( f(x) \)?
4. How can you find the points of discontinuity, if any, in \( f(x) \)?
5. What happens to \( f(x) \) as \( x \) approaches 2 from both the left and the right?

**Tip:** When evaluating piecewise functions, carefully check the intervals to ensure you select the correct piece of the function for your calculation.

Learn how to evaluate the piecewise function f(x) = |x - 4| at x = 12.5 with a step-by-step solution. Understand how to apply piecewise functions and absolute value functions. Suitable for students in Grades 10-12.

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