Math Problem Statement
Sketch a graph of the piecewise function f(x) = { -3 if x <= -2, x if -2 < x <= 2, 4 if x > 2 }.
Solution
The given piecewise function is defined as follows:
\begin{cases} -3 & \text{if } x \leq -2 \\ x & \text{if } -2 < x \leq 2 \\ 4 & \text{if } x > 2 \end{cases}$$ ### Steps to Sketch the Graph: 1. **For $$x \leq -2$$:** - The function is constant, $$f(x) = -3$$. - Plot a horizontal line at $$y = -3$$ for all $$x \leq -2$$. - Since $$f(x) = -3$$ is valid for $$x = -2$$ as well, use a **closed dot** at $$(-2, -3)$$. 2. **For $$-2 < x \leq 2$$:** - The function follows $$f(x) = x$$, which is a linear function with a slope of 1. - Plot this line for the range $$-2 < x \leq 2$$. - At $$x = -2$$, the function does not include the point, so use an **open dot** at $$(-2, -2)$$. - At $$x = 2$$, the function includes the point, so use a **closed dot** at $$(2, 2)$$. 3. **For $$x > 2$$:** - The function is constant, $$f(x) = 4$$. - Plot a horizontal line at $$y = 4$$ for all $$x > 2$$. - Use an **open dot** at $$(2, 4)$$, since $$f(x) = 4$$ does not include $$x = 2$$. ### Key Points to Include: - Horizontal line at $$y = -3$$ for $$x \leq -2$$, with a closed dot at $$(-2, -3)$$. - Line $$y = x$$ from $$x = -2$$ to $$x = 2$$, with an open dot at $$(-2, -2)$$ and a closed dot at $$(2, 2)$$. - Horizontal line at $$y = 4$$ for $$x > 2$$, with an open dot at $$(2, 4)$$. This should give you a clear graph of the piecewise function. Let me know if you want further details or clarifications! Here are some related questions: 1. What is the domain of the function? 2. How do you calculate the range of a piecewise function? 3. Can piecewise functions be continuous? 4. What is the significance of open and closed dots in graphing? 5. How can we identify if a piecewise function is differentiable? **Tip:** Always check for discontinuities at the break points of piecewise functions!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing Functions
Linear Functions
Constant Functions
Formulas
f(x) = -3 for x <= -2
f(x) = x for -2 < x <= 2
f(x) = 4 for x > 2
Theorems
None specific, but relates to the concept of piecewise-defined functions and their graphing.
Suitable Grade Level
Grades 8-10