Math Problem Statement
Solution
The problem asks to write the piecewise function whose graph is shown in the image.
Analysis of the Graph:
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Leftmost segment (for ): The line is sloped downwards and passes through the point and continues in the negative direction. This line has a slope of -1, indicating the equation should be for this interval.
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Middle segment (for ): The line passes through the point and continues until . This part of the graph looks like a straight line with slope 1. The equation for this segment is .
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Rightmost segment (for ): This is a quadratic-looking curve, and based on the shape and points it passes through, we can guess it's part of a downward parabola. But as it is not completely clear, we rely on the stated function for this interval.
The piecewise function is as follows:
\begin{cases} x - 2, & x = 2 \\ x + 2, & -2 \leq x \leq 0 \\ x - 2, & 0 \leq x \leq 2 \end{cases}$$ Let me know if you'd like more details or further explanation! Here are some follow-up questions related to piecewise functions: 1. What conditions must be true for a function to be continuous at a point? 2. How do you determine the equation of a line from two given points? 3. What is the general form of a quadratic equation, and how can you recognize it from a graph? 4. Can piecewise functions have discontinuities, and what types of discontinuities exist? 5. How do the domain and range of a piecewise function differ from a regular function? **Tip**: When sketching graphs, pay close attention to the slope and y-intercepts of the lines in each piece of the function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Equations
Graph Interpretation
Formulas
Slope-intercept form of a line: y = mx + b
Piecewise function structure: f(x) = { expression_1 for interval_1, expression_2 for interval_2, ... }
Theorems
Continuity of piecewise functions
Linear equations theorem
Suitable Grade Level
Grades 9-11