Math Problem Statement

Suppose that the function h is defined for all real numbers as follows: h(x) = 3 if x ≤ -2, h(x) = (x - 1)^2 + 2 if -2 < x < 2, h(x) = -3 if x ≥ 2. Find h(-2), h(0), and h(4).

Solution

We are tasked with evaluating the piecewise function h(x)h(x) defined as:

\begin{cases} 3 & \text{if } x \leq -2, \\ (x - 1)^2 + 2 & \text{if } -2 < x < 2, \\ -3 & \text{if } x \geq 2. \end{cases}$$ We need to calculate $$h(-2)$$, $$h(0)$$, and $$h(4)$$. --- ### Step-by-step Solution: 1. **Calculate $$h(-2)$$:** $$-2$$ falls under the condition $$x \leq -2$$. Thus: $$h(-2) = 3.$$ 2. **Calculate $$h(0)$$:** $$0$$ falls under the condition $$-2 < x < 2$$. Thus: $$h(0) = (0 - 1)^2 + 2 = (-1)^2 + 2 = 1 + 2 = 3.$$ 3. **Calculate $$h(4)$$:** $$4$$ falls under the condition $$x \geq 2$$. Thus: $$h(4) = -3.$$ --- ### Final Answers: - $$h(-2) = 3$$, - $$h(0) = 3$$, - $$h(4) = -3$$. Would you like a deeper explanation, or help with related problems? --- ### 5 Related Questions: 1. What is a piecewise function, and how are they used in mathematics? 2. How would $$h(x)$$ behave if the condition for $$-2 < x < 2$$ were modified? 3. Can you calculate $$h(1)$$ using the same function? 4. What does continuity mean in the context of piecewise functions? 5. How can we graph this piecewise function? --- ### Tip: When working with piecewise functions, carefully identify which condition the given $$x$$-value satisfies to avoid confusion.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Function Evaluation
Basic Algebra

Formulas

Piecewise function evaluation rules
(x - 1)^2 + 2

Theorems

Basic definition of piecewise functions

Suitable Grade Level

Grades 9-11