Math Problem Statement
Consider the function f defined piecewise as f(x): x^2 - 1 if -6 ≤ x ≤ -1; 2 - x if -1 < x < 2; -x^4 if 2 ≤ x ≤ 6. Find f(0).
Solution
The image shows a piecewise function defined as follows:
\begin{cases} x^2 - 1 & \text{if } -6 \leq x \leq -1 \\ 2 - x & \text{if } -1 < x < 2 \\ -x^4 & \text{if } 2 \leq x \leq 6 \end{cases}$$ The problem asks to find $$f(0)$$. To solve this, we need to determine which piece of the function applies for $$x = 0$$. - The first piece applies for $$-6 \leq x \leq -1$$, which does not include $$x = 0$$. - The second piece applies for $$-1 < x < 2$$, and $$0$$ lies within this interval. - The third piece applies for $$2 \leq x \leq 6$$, which does not include $$x = 0$$. Thus, $$f(0)$$ corresponds to the second piece of the function, which is $$f(x) = 2 - x$$. $$f(0) = 2 - 0 = 2$$ Therefore, $$f(0) = 2$$. Would you like further details or explanations on this solution? Here are 5 related questions you might find interesting: 1. How do you evaluate a piecewise function for different values of $$x$$? 2. What happens if $$x$$ lies exactly on the boundaries of two intervals in a piecewise function? 3. Can you graph the given piecewise function to visualize its behavior? 4. How can you determine the continuity of a piecewise function at its boundaries? 5. What is the significance of the piecewise function's different expressions over different intervals? **Tip:** When working with piecewise functions, always check which interval the input value belongs to before applying the function's rule.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Basic Algebra
Evaluation of Functions
Formulas
f(x) = x^2 - 1 (for -6 ≤ x ≤ -1)
f(x) = 2 - x (for -1 < x < 2)
f(x) = -x^4 (for 2 ≤ x ≤ 6)
Theorems
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Suitable Grade Level
Grades 9-12