Math Problem Statement
resuelve este ejercicio
Solution
To solve the exercise provided, we need to determine the domain, range, and graph of the piecewise function given by:
3 & , \ x \in (-7, -3] \\ 1 - 2x & , \ x \in (-2, 2) \\ |x - 4| - 6 & , \ x \in (2, 7] \end{cases}$$ ### 1. **Domain**: The domain of the function is the set of all $$x$$ values for which the function is defined. - The first piece $$f(x) = 3$$ is defined for $$x \in (-7, -3]$$. - The second piece $$f(x) = 1 - 2x$$ is defined for $$x \in (-2, 2)$$. - The third piece $$f(x) = |x - 4| - 6$$ is defined for $$x \in (2, 7]$$. **Overall Domain**: The union of all the intervals where each piece is defined, which is: $$\text{Domain} = (-7, -3] \cup (-2, 2) \cup (2, 7]$$ ### 2. **Range**: The range of the function is the set of all possible output values $$f(x)$$. - For $$x \in (-7, -3]$$, $$f(x) = 3$$. So, the output is always $$3$$. - For $$x \in (-2, 2)$$, $$f(x) = 1 - 2x$$. The function $$1 - 2x$$ is linear and decreases from $$1 - 2(-2) = 5$$ to $$1 - 2(2) = -3$$. - For $$x \in (2, 7]$$, $$f(x) = |x - 4| - 6$$. Since $$x \in (2, 7]$$, $$|x-4|$$ takes values between $$0$$ and $$3$$. Thus, $$|x - 4| - 6$$ ranges from $$-6$$ to $$-3$$. **Overall Range**: \[ \text{Range} = \{3\} \cup [-3, 5] \cup [-6, -3] \] Simplifying, the range is: $$\text{Range} = [-6, 5] \cup \{3\}$$ ### 3. **Graph**: To graph the function, we consider the three pieces separately: 1. **For** $$x \in (-7, -3]$$: The function is a constant, so it’s a horizontal line at $$y = 3$$. 2. **For** $$x \in (-2, 2)$$: The function is linear, so it’s a straight line with a negative slope that crosses the y-axis at $$y = 1$$. 3. **For** $$x \in (2, 7]$$: The function is a shifted absolute value function that starts from $$y = -4$$ at $$x = 4$$ and has a V-shape. Would you like to see the graph of this function or a more detailed explanation of any specific part? ### Follow-up Questions: 1. What is the significance of the domain in understanding the function? 2. How does the range change when altering the intervals in a piecewise function? 3. How do you determine whether a piecewise function is continuous? 4. Can the range of a piecewise function include isolated points? 5. How would the graph change if the intervals defining each piece were altered? ### Tip: When analyzing piecewise functions, it's essential to examine each piece separately to understand its behavior before considering the entire function's domain and range.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Domain and Range
Graphing Functions
Formulas
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Theorems
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Suitable Grade Level
Grades 10-12