Question
Study the function below and then answer the questions that follow.
\[
f(x)=\left\{\begin{array}{cl}
-x-2, & x<-2 \\
-x^{2}, & -2
Ask by Lowe Collins. in the United States
Feb 28,2025
Ask by Lowe Collins. in the United States
Feb 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The domain of \( f(x) \) is \( (-\infty, -2) \cup (-2, \infty) \).
Solution
The function is defined by three pieces:
• For x < -2: f(x) = -x - 2
• For -2 < x < 0: f(x) = -x²
• For x ≥ 0: f(x) = x
Notice that none of the pieces includes x = -2 (it’s excluded in both x < -2 and -2 < x < 0). Therefore, the function is defined for all real numbers except x = -2.
Thus, the domain of f(x) is: (-∞, -2) ∪ (-2, ∞).
This corresponds to the second option given.
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Simplify this solution Beyond the Answer
The function \( f(x) \) is defined for all real numbers except at the point \( x = -2 \), where the definition changes. So, the domain of \( f(x) \) encompasses all values of \( x \) except for \( -2 \). The correct way to represent the domain is \( (-\infty, -2) \cup (-2, \infty) \). That's like telling a tale of a map where every spot is treasure except for one: \( -2 \) just isn't on the menu! You've got the whole real number line to explore, minus that one little bump in the road. Happy studying!
