Math Problem Statement
Graph the piecewise function f(x) defined as:
f(x) =
x + 3 for -1 < x ≤ 2
-x + 9 for 2 < x < 6
on the axes provided.
Solution
The problem is asking you to graph the piecewise function defined as:
\begin{cases} x + 3 & \text{for } -1 < x \leq 2, \\ -x + 9 & \text{for } 2 < x < 6. \end{cases}$$ ### Step 1: Analyze Each Piece of the Function #### For $$x + 3$$ ($$-1 < x \leq 2$$): - This is a linear equation with slope $$1$$ and y-intercept $$3$$. - The domain is $$-1 < x \leq 2$$, meaning the graph starts just to the right of $$x = -1$$ and ends at $$x = 2$$ (inclusive). - At $$x = 2$$: $$f(2) = 2 + 3 = 5$$. - At $$x \to -1^+$$: $$f(x) = -1 + 3 = 2$$. Thus, this piece is a line segment from $$(-1, 2)$$ (open circle) to $$(2, 5)$$ (closed circle). #### For $$-x + 9$$ ($$2 < x < 6$$): - This is a linear equation with slope $$-1$$ and y-intercept $$9$$. - The domain is $$2 < x < 6$$, meaning the graph starts just to the right of $$x = 2$$ and ends just before $$x = 6$$. - At $$x \to 2^+$$: $$f(x) = -(2) + 9 = 7$$. - At $$x \to 6^-$$: $$f(x) = -(6) + 9 = 3$$. Thus, this piece is a line segment from $$(2, 7)$$ (open circle) to $$(6, 3)$$ (open circle). ### Step 2: Graphing Instructions 1. Plot the first piece ($$x + 3$$) as a line segment: - Start with an open circle at $$(-1, 2)$$. - Draw a straight line to $$(2, 5)$$ and place a closed circle there. 2. Plot the second piece ($$-x + 9$$) as a line segment: - Start with an open circle at $$(2, 7)$$. - Draw a straight line to $$(6, 3)$$, ending with an open circle. Would you like me to prepare a visual representation of this graph for you? ### Related Questions 1. What does the slope of each piece of the function represent? 2. Why is one endpoint open and the other closed for the first segment? 3. How do you handle overlapping endpoints in a piecewise function? 4. How does the graph change if the domains are modified? 5. What is the importance of specifying the domain for each piece? **Tip:** Always double-check the domain restrictions when graphing piecewise functions to ensure accuracy.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Linear equations
Graphing
Domain and range
Formulas
Slope-intercept form y = mx + b
Theorems
Linear function properties
Suitable Grade Level
Grades 9-12