Math Problem Statement
The function in the image is a piecewise function f(x) defined as follows:
f(x) =
{
3 - x , x < 2
3 , x = 2
-x^2 + 4 , x > 2
}
Solution
The function in the image is a piecewise function defined as follows:
\begin{cases} 3 - x & , \ x < 2 \\ 3 & , \ x = 2 \\ -x^2 + 4 & , \ x > 2 \end{cases}$$ This means: - For $$x < 2$$, the function is $$f(x) = 3 - x$$. - For $$x = 2$$, the function is a constant, $$f(x) = 3$$. - For $$x > 2$$, the function is a quadratic, $$f(x) = -x^2 + 4$$. Would you like more details about analyzing the behavior of this piecewise function, such as continuity or graphing? Here are five related questions you might explore: 1. Is $$f(x)$$ continuous at $$x = 2$$? 2. What is the domain and range of $$f(x)$$? 3. How would you graph this piecewise function? 4. What are the critical points and local extrema of $$f(x)$$? 5. How does the derivative of $$f(x)$$ behave in different intervals? **Tip:** When working with piecewise functions, always check for continuity and differentiability at the points where the function changes form.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Algebra
Quadratic Functions
Formulas
f(x) = 3 - x for x < 2
f(x) = 3 for x = 2
f(x) = -x^2 + 4 for x > 2
Theorems
Continuity of Piecewise Functions
Quadratic Function Properties
Suitable Grade Level
Grades 9-12