Math Problem Statement
Sketch the graph of the piecewise function and write the domain in interval notation: f(x) = {x + 3 if x < 2; -3x + 1 if x >= 2}.
Solution
To help you sketch the graph of the piecewise function and express the domain in interval notation, let's break down the function :
\begin{cases} x + 3 & \text{if } x < 2 \\ -3x + 1 & \text{if } x \geq 2 \end{cases}$$ ### Step-by-Step Breakdown: #### 1. **For $$x < 2$$ (Equation: $$f(x) = x + 3$$)** - This is a straight line with a slope of 1 and a y-intercept of 3. - The line will extend up to, but not including, $$x = 2$$. - At $$x = 2$$, the function does not include the value because the inequality is strict ($$x < 2$$), so we will draw an open dot at $$x = 2$$ on this part of the line. - **Example points**: - $$f(0) = 0 + 3 = 3$$ - $$f(1) = 1 + 3 = 4$$ - The function approaches $$f(2) = 2 + 3 = 5$$, but this value is not included (indicated by an open dot). #### 2. **For $$x \geq 2$$ (Equation: $$f(x) = -3x + 1$$)** - This is a straight line with a slope of -3 and a y-intercept of 1. - The line starts at $$x = 2$$ and extends to the right for $$x \geq 2$$. - At $$x = 2$$, the point is included since the inequality is non-strict ($$x \geq 2$$), so we place a closed dot at $$x = 2$$. - **Example points**: - $$f(2) = -3(2) + 1 = -6 + 1 = -5$$ - $$f(3) = -3(3) + 1 = -9 + 1 = -8$$ ### **Graphing Tips**: - Draw the first line $$f(x) = x + 3$$ for $$x < 2$$, ending with an open dot at $$(2, 5)$$. - Draw the second line $$f(x) = -3x + 1$$ for $$x \geq 2$$, starting with a closed dot at $$(2, -5)$$. ### **Domain in Interval Notation**: Since both parts of the function cover all real values of $$x$$, the domain is: $$\boxed{(-\infty, \infty)}$$ This means the function is defined for all real numbers. Would you like more details, or do you have any specific questions about this? Here are five related questions you might explore: 1. How can we find the range of the piecewise function? 2. What is the significance of the slope in each part of the piecewise function? 3. How does the function behave at $$x = 2$$? 4. Can we determine whether the function is continuous or discontinuous at $$x = 2$$? 5. How can the piecewise function be used to model real-world scenarios? **Tip**: When graphing piecewise functions, always check the boundary points carefully to decide whether to use open or closed dots!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Algebra
Graphing
Linear Equations
Formulas
f(x) = x + 3 for x < 2
f(x) = -3x + 1 for x >= 2
Theorems
-
Suitable Grade Level
Grades 10-12