Math Problem Statement
Sketch the graph of the following piecewise function: f(x) = {1 + x for x ≤ 3, 4 for x > 3}.
Solution
The function given in the image is a piecewise function:
\begin{cases} 1 + x & \text{for } x \leq 3 \\ 4 & \text{for } x > 3 \end{cases}$$ ### Steps to sketch the graph: 1. **For $$x \leq 3$$, the function is $$f(x) = 1 + x$$:** - This is a straight line with a slope of 1 and a y-intercept of 1. - The line extends up to $$x = 3$$. At $$x = 3$$, the value of $$f(x) = 1 + 3 = 4$$. 2. **For $$x > 3$$, the function is $$f(x) = 4$$:** - This is a horizontal line at $$y = 4$$ starting from $$x = 3$$ (not including 3) and going to infinity for increasing $$x$$. 3. **Graph behavior at $$x = 3$$:** - At $$x = 3$$, both the line from the first part and the constant line from the second part meet at $$f(3) = 4$$. Therefore, there is no discontinuity or break at this point, making the graph continuous. ### Key points to consider: - From $$x = -\infty$$ to $$x = 3$$, the graph is a straight line increasing with slope 1. - For $$x > 3$$, the graph becomes a horizontal line at $$y = 4$$. - The graph is continuous at $$x = 3$$ because both parts of the piecewise function agree at that point. Would you like me to generate a sketch of the graph based on this explanation? --- Here are 5 questions to further explore this problem: 1. How would the graph change if the second part of the piecewise function was $$f(x) = 5$$ for $$x > 3$$? 2. What happens to the graph if we introduce a discontinuity at $$x = 3$$ (e.g., change $$f(3)$$)? 3. How do we determine the slope of the linear part of the graph based on the equation $$f(x) = 1 + x$$? 4. What role does the boundary $$x = 3$$ play in defining the overall shape of the graph? 5. How would we describe the domain and range of this function? ### Tip: When sketching piecewise functions, always check for continuity at the points where the function definition changes.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Functions
Graphing
Formulas
f(x) = 1 + x for x ≤ 3
f(x) = 4 for x > 3
Theorems
Continuity of Piecewise Functions
Suitable Grade Level
Grades 10-12