Math Problem Statement
Given the piecewise function:
y =
\begin{cases}
x^2 & \text{if } x < 1 \\
(x-1)^2 & \text{if } 1 \leq x \leq 2 \\
3 - x & \text{if } x > 2
\end{cases}
Explain the behavior of the function.
Solution
The given equation is a piecewise function, where is defined differently for three intervals of :
\begin{cases} x^2 & \text{if } x < 1 \\ (x-1)^2 & \text{if } 1 \leq x \leq 2 \\ 3 - x & \text{if } x > 2 \end{cases}$$ ### Explanation of Each Piece: 1. **For $$x < 1$$: $$y = x^2$$** This is a quadratic equation where the graph is a parabola opening upwards. 2. **For $$1 \leq x \leq 2$$: $$y = (x-1)^2$$** This is also a quadratic equation, but it's shifted to the right by 1 unit (vertex at $$x = 1$$, $$y = 0$$). 3. **For $$x > 2$$: $$y = 3 - x$$** This is a linear equation with a negative slope of -1, starting at $$y = 1$$ when $$x = 2$$. --- Would you like a graph of this function or a more detailed analysis, such as continuity and differentiability? --- ### Follow-up Questions: 1. What does it mean for a function to be continuous at $$x = 1$$ or $$x = 2$$? 2. How do you check if the function is differentiable at $$x = 1$$ and $$x = 2$$? 3. What is the range of this piecewise function? 4. Can you identify any critical points or local extrema for this function? 5. What would the graph of this function look like, and how would the pieces connect? **Tip:** When analyzing piecewise functions, always check for **continuity** and **differentiability** at the transition points!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Quadratic Functions
Linear Functions
Continuity
Differentiability
Formulas
y = x^2 for x < 1
y = (x - 1)^2 for 1 ≤ x ≤ 2
y = 3 - x for x > 2
Theorems
Continuity of a function at a point
Differentiability of a function at a point
Suitable Grade Level
Grades 9-11
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