Math Problem Statement
Solve
Solution
The image shows a piecewise function defined as follows:
x + 1 & \text{if } x \leq -1 \\ x^2 - 1 & \text{if } -1 < x \leq 1 \\ x - 1 & \text{if } x > 1 \end{cases}$$ Let's go through the process of sketching the graph of this function. ### Step-by-Step Graphing 1. **For $$x \leq -1$$:** - The function is $$f(x) = x + 1$$. - This is a linear function with a slope of 1 and a y-intercept of 1. - At $$x = -1$$, $$f(-1) = -1 + 1 = 0$$. This is a closed point since the inequality is $$x \leq -1$$. - Draw this line for $$x \leq -1$$, starting from the point $$(-1, 0)$$ and extending to the left. 2. **For $$-1 < x \leq 1$$:** - The function is $$f(x) = x^2 - 1$$. - This is a quadratic function, a parabola opening upwards, shifted down by 1 unit. - At $$x = -1$$, $$f(-1) = (-1)^2 - 1 = 0$$ (but this point is not included because of the strict inequality). - At $$x = 0$$, $$f(0) = 0^2 - 1 = -1$$. - At $$x = 1$$, $$f(1) = 1^2 - 1 = 0$$. This is a closed point because of the inequality $$\leq 1$$. - Plot this parabola between $$x = -1$$ and $$x = 1$$, with the corresponding points at $$(-1, 0)$$ (open), $$(0, -1)$$, and $$(1, 0)$$ (closed). 3. **For $$x > 1$$:** - The function is $$f(x) = x - 1$$. - This is a linear function with a slope of 1 and a y-intercept of -1. - At $$x = 1$$, $$f(1) = 1 - 1 = 0$$, but this point is not included (open circle at $$(1, 0)$$). - Draw this line for $$x > 1$$, starting just after $$x = 1$$ and extending to the right. ### Final Graph: - The graph consists of three segments: - A line starting at $$(-1, 0)$$ and going leftwards. - A parabola from $$x = -1$$ (not including $$x = -1$$) to $$x = 1$$ (including $$x = 1$$). - A line starting from just after $$x = 1$$ and moving rightwards. This graph will illustrate the behavior of the piecewise function across different intervals. Would you like more details on any specific part of the graph? Or perhaps have further questions about this function? Here are some related questions you might want to explore: 1. How do the limits from the left and right at $$x = -1$$ and $$x = 1$$ compare? 2. What is the derivative of each piece of the function, and what does it tell us about the slope? 3. How does the graph change if we modify the constants in each piece? 4. Can you identify any points of discontinuity in the function? 5. How would you find the area between the curve and the x-axis within specific intervals? **Tip:** When dealing with piecewise functions, always carefully check the conditions at the boundary points to ensure you correctly interpret the behavior of the function at those points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Graphing
Formulas
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Theorems
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Suitable Grade Level
Grades 10-12