Math Problem Statement
The function f(x) is to be graphed on a coordinate plane.
f(x) =
-x, x < 0
1, x ≥ 0
At what point should an open circle be drawn?
(-1, 0)
(0, 0)
(0, 1)
(1, 0)
Solution
The question involves a piecewise function defined as:
\begin{cases} -x, & x < 0 \\ 1, & x \geq 0 \end{cases}$$ The problem asks at which point an open circle should be drawn when graphing the function on a coordinate plane. Here's how we analyze it: 1. For $$x < 0$$, the function $$f(x) = -x$$ is valid. 2. For $$x \geq 0$$, the function $$f(x) = 1$$ is valid. 3. At $$x = 0$$, there is a transition from the first case ($$-x$$) to the second case ($$1$$). Since the domain boundary is at $$x = 0$$, we evaluate the behavior: - From the first case ($$f(x) = -x$$): At $$x = 0$$, $$f(x) = -0 = 0$$. This value would need an open circle because this part is **not included** when $$x = 0$$. - From the second case ($$f(x) = 1$$): At $$x = 0$$, $$f(x) = 1$$. This value is **included** because the inequality is $$x \geq 0$$. Thus, an **open circle** is drawn at $$(0, 0)$$ to indicate that this value is excluded from the function. The correct answer is: **$$(0, 0)$$.** --- Would you like a more detailed explanation or visual guidance on graphing this function? Here are some related questions for practice: 1. How do you graph a piecewise function with multiple intervals? 2. How do open and closed circles on a graph relate to inequalities? 3. What are the key steps to determine where a function is discontinuous? 4. How would the graph of $$f(x) = -x$$ look for $$x < 0$$? 5. What are common mistakes to avoid when graphing piecewise functions? **Tip**: Always carefully check the boundary points for inclusion ($$\leq, \geq$$) or exclusion ($$<, >$$) when dealing with piecewise functions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing Functions
Domain and Range
Formulas
f(x) = -x for x < 0
f(x) = 1 for x ≥ 0
Theorems
Definition of Piecewise Functions
Behavior of Graphs at Boundary Points
Suitable Grade Level
Grades 9-11