Math Problem Statement
Given the piecewise defined function:f(x)={x−2ifx≤2−2x+3ifx>2f(x)={x-2ifx≤2-2x+3ifx>2
Solution
Let's analyze the piecewise defined function :
\begin{cases} x - 2 & \text{if } x \leq 2 \\ -2x + 3 & \text{if } x > 2 \end{cases}$$ ### Step-by-Step Analysis 1. **For $$x \leq 2$$:** The function is defined as $$f(x) = x - 2$$. This is a linear equation with a slope of 1 and a y-intercept of -2. It will be a straight line for all values of $$x$$ less than or equal to 2. 2. **For $$x > 2$$:** The function is defined as $$f(x) = -2x + 3$$. This is also a linear equation but with a slope of -2 and a y-intercept of 3. It will be a straight line for all values of $$x$$ greater than 2. ### Points of Interest - At $$x = 2$$, we need to check the value of both pieces to ensure continuity or identify a jump. - For $$x \leq 2$$: $$f(2) = 2 - 2 = 0$$ - For $$x > 2$$: We check the value as we approach from the right side. $$\lim_{x \to 2^+} f(x) = -2(2) + 3 = -4 + 3 = -1$$ ### Conclusion: - **At $$x = 2$$,** the function $$f(x) = x - 2$$ gives $$f(2) = 0$$. - As $$x$$ approaches 2 from the right, $$f(x) = -2x + 3$$ gives $$-1$$. Thus, there is a **jump discontinuity** at $$x = 2$$ since the left-hand limit ($$0$$) and right-hand limit ($$-1$$) do not match. ### Graphical Interpretation: - **For $$x \leq 2$$:** Plot the line $$y = x - 2$$ up to $$x = 2$$ inclusive. This line passes through points $$(0, -2)$$ and $$(2, 0)$$. - **For $$x > 2$$:** Plot the line $$y = -2x + 3$$ starting just after $$x = 2$$. This line starts just after $$(2, -1)$$ and has a negative slope. ### Graph Behavior: - The graph will have a continuous line segment from $$x = -\infty$$ to $$x = 2$$ (inclusive) following $$y = x - 2$$. - After $$x = 2$$, the graph jumps to $$(2, -1)$$ and continues along the line $$y = -2x + 3$$. Would you like further details or have any questions? Here are 5 related questions for further practice: 1. What is the range of $$f(x)$$ given this piecewise function? 2. How would the graph change if the condition was $$x < 2$$ instead of $$x \leq 2$$? 3. Determine the points of intersection for both linear parts of the piecewise function. 4. How do you determine if a piecewise function is continuous at a point? 5. What changes if the second piece is $$f(x) = -2x + 5$$ instead of $$f(x) = -2x + 3$$? **Tip:** Always check the endpoint and transition points in piecewise functions to determine continuity and to find exact function values.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Equations
Continuity
Formulas
-
Theorems
Definition of Continuity
Suitable Grade Level
Grades 11-12
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