Math Problem Statement
Graph the function f(x) = {x + 4 for x > 1, x - 3 for x <= 1} and state the domain and range.
Solution
The given piecewise function is:
\begin{cases} x + 4 & \text{for } x > 1 \\ x - 3 & \text{for } x \leq 1 \end{cases}$$ ### Step 1: Analyze the function 1. For $$x > 1$$, the function is $$f(x) = x + 4$$, which is a linear equation with slope $$1$$ and y-intercept $$4$$. 2. For $$x \leq 1$$, the function is $$f(x) = x - 3$$, which is a linear equation with slope $$1$$ and y-intercept $$-3$$. The graph transitions at $$x = 1$$: - For $$x > 1$$, the graph is an open circle at $$x = 1$$ because the inequality is strict ($$x > 1$$). - For $$x \leq 1$$, the graph includes $$x = 1$$, represented by a closed circle. ### Step 2: Determine the domain and range - **Domain:** The function is defined for all real numbers ($$-\infty, \infty$$). - **Range:** Since both linear components ($$x + 4$$ and $$x - 3$$) cover all real values as $$x$$ moves through the domain, the range is also $$(-\infty, \infty)$$. ### Step 3: Verify the graph The correct graph is **B**: - The left part ($$x - 3$$) is a line with a closed circle at $$x = 1$$. - The right part ($$x + 4$$) is a line with an open circle at $$x = 1$$. ### Final Answer: - **Domain:** $$(-\infty, \infty)$$ - **Range:** $$(-\infty, \infty)$$ Would you like me to explain more about piecewise functions? --- ### 5 Related Questions: 1. What are some common methods for graphing piecewise functions? 2. How do we find the intersection points of two linear pieces in a piecewise function? 3. Can piecewise functions have discontinuities? How can we find them? 4. How do domain restrictions affect the graph of a piecewise function? 5. What are some real-world applications of piecewise functions? ### Tip: When analyzing a piecewise function, always check for continuity at the boundaries of each piece. This ensures accurate graphing and interpretation!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Equations
Domain and Range
Formulas
f(x) = x + 4 (for x > 1)
f(x) = x - 3 (for x <= 1)
Theorems
Definition of domain and range in functions
Suitable Grade Level
Grades 9-11