Math Problem Statement
f(x) = {x − 2, if x ≤ 0 2x+1, ifx>0
Solution
To understand the function , we need to break it down according to its piecewise definition:
\begin{cases} x - 2 & \text{if } x \leq 0 \\ 2x + 1 & \text{if } x > 0 \end{cases}$$ Let's analyze each part separately. ### For $$x \leq 0$$: When $$x$$ is less than or equal to zero, the function is defined as: $$f(x) = x - 2$$ ### For $$x > 0$$: When $$x$$ is greater than zero, the function is defined as: $$f(x) = 2x + 1$$ ### Examples: 1. If $$x = -1$$: Since $$-1 \leq 0$$, we use the first part of the function: $$f(-1) = -1 - 2 = -3$$ 2. If $$x = 2$$: Since $$2 > 0$$, we use the second part of the function: $$f(2) = 2(2) + 1 = 4 + 1 = 5$$ 3. If $$x = 0$$: Since $$0 \leq 0$$, we use the first part of the function: $$f(0) = 0 - 2 = -2$$ Would you like a graphical representation of this function or more detailed information about how it behaves for different values of $$x$$? --- ### Questions for Further Exploration: 1. How does the function $$f(x)$$ change when $$x$$ transitions from negative to positive? 2. What is the value of $$f(x)$$ at $$x = 0$$? 3. How would you describe the behavior of $$f(x)$$ as $$x$$ approaches 0 from the left? 4. How would you describe the behavior of $$f(x)$$ as $$x$$ approaches 0 from the right? 5. What are the intercepts of $$f(x)$$ with the x-axis and y-axis? 6. How would you plot $$f(x)$$ on a coordinate plane? 7. Can you find the domain and range of $$f(x)$$? 8. How would the function change if the conditions for $$x$$ were altered? ### Tip: When dealing with piecewise functions, carefully analyze each segment of the function separately and consider how they connect at the boundaries.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Formulas
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Theorems
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Suitable Grade Level
Grades 9-12