Math Problem Statement
Find the domain and the range of the function:
f(x) = {1/2 x, for x < 0; x + 3, for x >= 0}
Solution
Let's analyze the piecewise-defined function provided in the image:
\begin{cases} \frac{1}{2}x, & \text{for } x < 0 \\ x + 3, & \text{for } x \geq 0 \end{cases}$$ ### 1. **Domain of the function**: - The first piece, $$\frac{1}{2}x$$, is valid for $$x < 0$$. - The second piece, $$x + 3$$, is valid for $$x \geq 0$$. - Therefore, the domain of $$f(x)$$ is all real numbers, $$(-\infty, \infty)$$, since there is no restriction on the values $$x$$ can take. This matches option **B**: $$(-\infty, \infty)$$. ### 2. **Range of the function**: - For $$x < 0$$, $$\frac{1}{2}x$$ produces values from $$(-\infty, 0)$$. - For $$x \geq 0$$, $$x + 3$$ produces values starting from $$3$$ and increasing to $$(\infty)$$. - Combining these two parts, the range is $$(-\infty, 3) \cup [3, \infty)$$. This matches option **A**: $$(-\infty, 0)$$ and $$[3, \infty)$$. ### Conclusion: - **Domain**: Option **B**: $$(-\infty, \infty)$$ - **Range**: Option **A**: $$(-\infty, 0)$$ and $$[3, \infty)$$ Would you like more details on any part of this explanation or have any questions? Here are some related questions: 1. What is the general approach to finding the domain of piecewise functions? 2. How can we determine the range of a function from its equation? 3. Why is the range split into two parts in this case? 4. How does the slope of $$\frac{1}{2}x$$ affect the function for $$x < 0$$? 5. How does the value of $$x+3$$ behave for larger values of $$x$$? **Tip**: When dealing with piecewise functions, carefully analyze each segment separately before combining the results for the domain and range.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Algebra
Functions
Formulas
f(x) = {1/2 x for x < 0; x + 3 for x >= 0}
Theorems
Domain and Range Analysis
Suitable Grade Level
Grades 9-12
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