Math Problem Statement
Find the domain and range of the function 1 1 0 1 1 1 1 ( ) x if x if x x if x f x (2 Marks)
Solution
Let's analyze the piecewise function given:
-x - 1 & \text{if } x < -1 \\ x^2 & \text{if } -1 \leq x \leq 1 \\ x + 1 & \text{if } x > 1 \end{cases}$$ ### Domain: The domain of a function is the set of all possible input values (x-values) for which the function is defined. - For the first piece, $$-x - 1$$, there is no restriction on $$x$$ other than $$x < -1$$. - For the second piece, $$x^2$$, there is no restriction on $$x$$ other than $$-1 \leq x \leq 1$$. - For the third piece, $$x + 1$$, there is no restriction on $$x$$ other than $$x > 1$$. Since these three pieces cover all real numbers, the domain of $$f(x)$$ is all real numbers: $$\text{Domain: } (-\infty, \infty)$$ ### Range: The range of a function is the set of all possible output values (y-values). - For $$x < -1$$, $$f(x) = -x - 1$$. As $$x$$ approaches $$-1$$ from the left, $$f(x)$$ approaches 0. As $$x$$ decreases towards $$-\infty$$, $$f(x)$$ increases towards $$\infty$$. Thus, the range for this piece is $$(-\infty, 0)$$. - For $$-1 \leq x \leq 1$$, $$f(x) = x^2$$. The minimum value occurs at $$x = 0$$, giving $$f(x) = 0$$. The maximum value occurs at $$x = \pm1$$, giving $$f(x) = 1$$. Thus, the range for this piece is $$[0, 1]$$. - For $$x > 1$$, $$f(x) = x + 1$$. As $$x$$ increases from 1, $$f(x)$$ increases from 2 onwards. Thus, the range for this piece is $$(2, \infty)$$. By combining the ranges of the three pieces, we get the overall range: $$\text{Range: } (-\infty, 0) \cup [0, 1] \cup (2, \infty)$$ ### Summary: - **Domain:** $$(-∞, ∞)$$ - **Range:** $$(-∞, 0) \cup [0, 1] \cup (2, ∞)$$ Would you like more details or have any questions? ### Related Questions: 1. How do you graph piecewise functions? 2. What are the different methods to find the range of a function? 3. How does the domain of a function differ from its range? 4. Can a function have a discontinuous range? 5. What are the implications of a function being defined as piecewise? 6. How do you handle inequalities when determining the domain and range? 7. Can a piecewise function be continuous? 8. How can the inverse of a piecewise function be determined? ### Tip: When determining the range of a piecewise function, it's helpful to consider the range of each piece separately and then combine them, checking for any overlaps or gaps.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Domain and Range
Formulas
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Theorems
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Suitable Grade Level
Grades 11-12