Math Problem Statement
Quiz:Chapter 1 Skills Check Quiz (Integrated Review) Question 1 of 6 This quiz: 6 point(s) possible This question: 1 point(s) possible
Skip to Main content Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question content area top Part 1 Graph f(x). Start 1 By 2 Matrix 1st Row 1st Column f left parenthesis x right parenthesis equals 2nd Column left brace Start 3 By 2 Matrix 1st Row 1st Column 2 x comma 2nd Column for x less than 0 2nd Row 1st Column negative 3 comma 2nd Column for x equals 0 3rd Row 1st Column x minus 2 comma 2nd Column for x greater than 0 EndMatrix EndMatrix
Question content area bottom Part 1 Choose the correct graph below. A. -6 6 -6 6 x y
A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. A line that rises from left to right passes through the points (negative 2, negative 4) and (2, 4). An open circle is plotted at (0, 0) on the line. A solid dot is plotted at (0, negative 3). B. -6 6 -6 6 x y
A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. A graph has three parts. The first part is a ray that falls from right to left, starting at an open circle at (0, 0) and passing through (negative 1, negative 2). The second part is a solid dot at (0, negative 3). The third part is a ray that rises from left to right, starting at an open circle at (0, negative 2) and passing through (1, negative 1). C. -6 6 -6 6 x y
A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. A graph has three parts. The first part is a ray that falls from right to left, starting at an open circle at (0, negative 2) and passing through (negative 1, negative 3). The second part is a solid dot at (0, negative 3). The third part is a ray that rises from left to right, starting at an open circle at (0, 0) and passing through (1, 2). D. -6 6 -6 6 x y
A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. A graph has three parts. The first part is a ray that falls from right to left, starting at an open circle at (0, 0) and passing through (negative 1, negative 2). The second part is a solid dot at (0, negative 3). The third part is a ray that rises from left to right, starting at an open circle at (0, 2) and passing through (1, 3). Choose the correct graph below.
Solution
We are tasked with graphing the piecewise function:
\begin{cases} 2x & \text{for } x < 0 \\ -3 & \text{for } x = 0 \\ x - 2 & \text{for } x > 0 \end{cases}$$ ### Step-by-Step Breakdown: 1. **For $$x < 0$$, the function is $$f(x) = 2x$$:** - This is a straight line with slope 2, passing through the origin (0, 0). However, for $$x < 0$$, this line will only be valid for negative values of $$x$$. - At $$x = 0$$, $$f(x) = 2(0) = 0$$, but we will mark this with an open circle because the function does not include $$x = 0$$ for this part. 2. **For $$x = 0$$, the function is $$f(x) = -3$$:** - At $$x = 0$$, the function takes the value $$f(0) = -3$$, which we represent with a solid dot at $$(0, -3)$$. 3. **For $$x > 0$$, the function is $$f(x) = x - 2$$:** - This is another straight line, but with slope 1 and y-intercept at $$-2$$. This line is valid only for positive values of $$x$$. - At $$x = 0$$, $$f(0) = 0 - 2 = -2$$, but since $$x = 0$$ is not included for this part, we mark this point with an open circle. ### Key features of the graph: - An open circle at $$(0, 0)$$ for the first part $$f(x) = 2x$$ when $$x < 0$$. - A solid dot at $$(0, -3)$$ for $$f(x) = -3$$ when $$x = 0$$. - An open circle at $$(0, -2)$$ and a line starting from this point with a slope of 1 for $$f(x) = x - 2$$ when $$x > 0$$. ### The correct graph is: - **Option B**: This graph matches the behavior of the piecewise function described, with the correct locations of open circles, solid dots, and lines for each part of the function. Let me know if you want more details or have any questions. --- Here are five related questions: 1. How do you graph piecewise functions with more complex conditions? 2. What is the general method for finding intercepts of piecewise functions? 3. How do you determine the domain and range of a piecewise function? 4. What is the importance of open and closed circles in graphing functions? 5. How do you handle discontinuities in piecewise functions? **Tip:** When graphing piecewise functions, always plot the boundary points carefully, especially considering whether they should be open or closed circles.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing
Linear Equations
Formulas
f(x) = 2x for x < 0
f(x) = -3 for x = 0
f(x) = x - 2 for x > 0
Theorems
Continuity and Discontinuity in Functions
Graphing Piecewise Functions
Suitable Grade Level
Grades 9-11