Consider the following piecewise-defined function.

f(x)=⎧⎩⎨⎪⎪⎪⎪−2x−22x2+2x34x−1ifx≤−5if−5<x≤5ifx>5f(x)={−2x−2ifx≤−52x2+2xif−5<x≤534x−1ifx>5

Step 2 of 3 :  

Evaluate this function at x=−5x=−5. Express your answer as an integer or simplified fraction. If the function is undefined at the given value, indicate "Undefined".

To evaluate the piecewise-defined function at \( x = -5 \), we need to determine which piece of the function applies for \( x = -5 \).

The given function is:
\[
f(x) =
\begin{cases}
-2x - 2, & \text{if } x \leq -5 \\
2x^2 + 2x, & \text{if } -5 < x \leq 5 \\
34x - 1, & \text{if } x > 5
\end{cases}
\]

Since \( x = -5 \), we use the first piece of the function, \( f(x) = -2x - 2 \).

Now, substitute \( x = -5 \) into the expression:
\[
f(-5) = -2(-5) - 2 = 10 - 2 = 8.
\]

Thus, the value of the function at \( x = -5 \) is \( 8 \).

Would you like more details or have any questions?

Here are five related questions to explore further:
1. What is the value of \( f(x) \) at \( x = 0 \)?
2. How would you evaluate \( f(x) \) at \( x = 6 \)?
3. What is the graph of this piecewise function?
4. How can we determine if a piecewise function is continuous at a given point?
5. What happens to the function when \( x \) is near the boundary points?

**Tip:** When evaluating piecewise functions, always check which interval your input belongs to in order to use the correct expression.

Consider the following piecewise-defined function. f(x) = { -2x - 2 if x ≤ -5, 2x^2 + 2x if -5 < x ≤ 5, 34x - 1 if x > 5 }. Evaluate this function at x = -5. 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-defined function at x = -5. This problem involves understanding function evaluation within specific intervals and covers key concepts in piecewise functions. Suitable for high school students in Grades 10-12.

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