Math Problem Statement
do the questions in this. i don't want any latex so i can copy paste it:
Graph the piecewise function given by h(x) = (x + 3, x < -2), (x^2, -2 ≤ x < 1), (-x + 2, x ≥ 1) on the coordinate plane.
Solution
Let's break down and solve the problems from the image step by step.
1. First Piecewise Function :
We are given the following piecewise function:
-x^2 + 4 & x < 1 \\ 2x - 1 & x \geq 1 \end{cases}$$ **Explanation:** - For $$x < 1$$, we use the expression $$-x^2 + 4$$. - For $$x \geq 1$$, we use the expression $$2x - 1$$. We also see the graph already provided for this piecewise function. Important points from the graph: - For $$x = 1$$, the function $$-x^2 + 4$$ has an open circle (not included) because $$x = 1$$ is not in the domain of this part of the function. - For $$x = 1$$, the function $$2x - 1$$ has a filled circle (included) because $$x = 1$$ is part of this domain. ### 2. Second Piecewise Function $$h(x)$$: We are asked to graph the following piecewise function: $$h(x) = \begin{cases} x + 3 & x < -2 \\ x^2 & -2 \leq x < 1 \\ -x + 2 & x \geq 1 \end{cases}$$ **Step-by-step instructions for graphing:** 1. **For $$x < -2$$, use $$h(x) = x + 3$$**: - This is a linear function with a slope of 1 and a y-intercept of 3. - The graph is a straight line, and at $$x = -2$$, we will have an open circle at the point $$(-2, 1)$$. 2. **For $$-2 \leq x < 1$$, use $$h(x) = x^2$$**: - This is a quadratic function (a parabola). - At $$x = -2$$, we have the point $$(-2, 4)$$, and since the domain includes $$x = -2$$, it will be a filled circle. - The graph of $$x^2$$ continues until $$x = 1$$, where the point is $$(1, 1)$$ but with an open circle. 3. **For $$x \geq 1$$, use $$h(x) = -x + 2$$**: - This is another linear function with a slope of -1 and a y-intercept of 2. - The graph starts at $$x = 1$$ (with a filled circle at $$(1, 1)$$) and continues downward. Now that we've broken down the functions and the steps, you can proceed by plotting each piece on the grid provided in the image. --- Would you like a more detailed breakdown of the graphing process, or do you have any questions on this? ### 5 Related Questions: 1. What is the difference between an open circle and a filled circle in a piecewise function graph? 2. How do you identify the correct domain for each expression in a piecewise function? 3. Can a piecewise function have discontinuities, and if so, how do they appear on the graph? 4. How can we check if a piecewise function is continuous at a particular point? 5. What is the importance of domain restrictions in a piecewise function? ### Tip: Always label key points, like open and closed circles, clearly on your graph to avoid confusion about which function is active at each point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Domain and range
Graphing linear functions
Graphing quadratic functions
Formulas
Linear equation: y = mx + b
Quadratic equation: y = ax^2 + bx + c
Theorems
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Suitable Grade Level
Grades 9-11