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Evaluate this function at x=−2. Express your answer as an integer or simplified fraction. If the function is undefined at the given value, indicate "Undefined".

To evaluate the function \( f(x) \) at \( x = -2 \), we'll use the piecewise definition provided in the image:

\[
f(x) = 
\begin{cases} 
-\frac{1}{4}x - 6 & \text{if } x \leq -2 \\
-3x - 3 & \text{if } x > -2
\end{cases}
\]

Since \( x = -2 \), we need to use the first part of the piecewise function because \( -2 \leq -2 \).

Substituting \( x = -2 \) into the function:

\[
f(-2) = -\frac{1}{4}(-2) - 6
\]

Calculate the expression step by step:

\[
f(-2) = \frac{1}{2} - 6 = \frac{1}{2} - \frac{12}{2} = \frac{1 - 12}{2} = \frac{-11}{2}
\]

Thus, the value of the function at \( x = -2 \) is \( \frac{-11}{2} \).

Would you like more details or have any questions? Here are some related questions you might consider:

1. What is the value of \( f(x) \) at \( x = 0 \)?
2. How would you find the derivative of \( f(x) \) for \( x \leq -2 \)?
3. What is the domain of the function \( f(x) \)?
4. How does the function behave as \( x \) approaches \( -2 \) from the left and the right?
5. Is the function continuous at \( x = -2 \)? Why or why not?

**Tip:** When working with piecewise functions, always pay attention to the specific condition for each piece before substituting the value of \( x \).

Evaluate this function at x = −2. Express your answer as an integer or simplified fraction. If the function is undefined at the given value, indicate 'Undefined'.

Learn how to evaluate a piecewise function at x = -2 with a detailed step-by-step solution. This problem covers key concepts in algebra, including piecewise functions and linear equations. Suitable for students in Grades 9-11.

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