Math Problem Statement
State the domain and range of f.
Solution
The function is defined as follows:
1 - x^2 & \text{if } x \leq 0 \\ 2x + 1 & \text{if } x > 0 \end{cases}$$ ### **Domain:** The domain of a function is the set of all possible input values (x-values) for which the function is defined. - For the first piece $$1 - x^2$$, the function is defined for $$x \leq 0$$. - For the second piece $$2x + 1$$, the function is defined for $$x > 0$$. Combining these, the domain of $$f(x)$$ is all real numbers: $$\text{Domain of } f(x): \ (-\infty, \infty)$$ ### **Range:** The range of a function is the set of all possible output values (y-values) that the function can produce. - For the first piece $$1 - x^2$$, as $$x$$ takes values less than or equal to 0, the maximum value of $$f(x)$$ occurs at $$x = 0$$, giving $$f(0) = 1 - 0^2 = 1$$. As $$x$$ decreases, $$f(x)$$ decreases without bound, so the range is $$(-\infty, 1]$$. - For the second piece $$2x + 1$$, since $$x > 0$$, $$f(x)$$ increases without bound as $$x$$ increases. The minimum value occurs as $$x$$ approaches 0 from the right, giving $$f(x)$$ values just slightly greater than 1. Thus, the range is $$(1, \infty)$$. Combining these, the range of $$f(x)$$ is: $$\text{Range of } f(x): \ (-\infty, \infty)$$ ### Summary: - **Domain:** $$(-\infty, \infty)$$ - **Range:** $$(-\infty, \infty)$$ Do you want details or have any questions? Here are 5 related questions to further explore the topic: 1. How would the domain and range change if the condition on $$x$$ were modified to $$x < 0$$ and $$x \geq 0$$? 2. What is the graph of $$f(x)$$ based on the given piecewise function? 3. Can you find the points of discontinuity for $$f(x)$$? 4. How would you determine if $$f(x)$$ is a continuous function? 5. How do the different parts of the piecewise function interact at the boundary $$x = 0$$? **Tip:** When determining the range of a piecewise function, always consider the maximum and minimum values that each piece can achieve within its specified interval.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 10-12