Math Problem Statement
Suppose that the function g is defined, for all real numbers, as follows:
g(x) =
{
1 if x ≤ -2
x if -2 < x ≤ 2
1 if x > 2
}
Graph the function g.
Solution
The problem you provided involves graphing a piecewise-defined function, which is given as follows:
\begin{cases} 1 & \text{if } x \leq -2 \\ x & \text{if } -2 < x \leq 2 \\ 1 & \text{if } x > 2 \end{cases}$$ ### To graph this piecewise function: 1. **For $$x \leq -2$$**: The function is constant and equal to 1. So for all values of $$x \leq -2$$, the graph will be a horizontal line at $$y = 1$$. 2. **For $$-2 < x \leq 2$$**: The function is equal to $$x$$, which means the graph will be the line $$y = x$$ for this interval. This is a diagonal line passing through the origin, but restricted between $$x = -2$$ and $$x = 2$$. 3. **For $$x > 2$$**: The function is again constant and equal to 1, so for all values of $$x > 2$$, the graph will be a horizontal line at $$y = 1$$. ### Key points to plot: - **At $$x = -2$$**: The value is $$g(-2) = 1$$. So, there is a closed circle at $$(-2, 1)$$. - **At $$x = 2$$**: The value is $$g(2) = 2$$. So, there is a closed circle at $$(2, 2)$$. - **For $$x > 2$$** and $$x \leq -2$$**: The graph is a flat line at $$y = 1$$ on both sides. ### Graphing Strategy: - Plot a horizontal line at $$y = 1$$ for $$x \leq -2$$ and $$x > 2$$. - Plot a diagonal line with slope 1 between $$x = -2$$ and $$x = 2$$. - Make sure to use solid circles at the transition points for continuous functions, such as at $$x = -2$$ and $$x = 2$$. Would you like a detailed plot or any further explanation? Here are 5 follow-up questions: 1. How do you determine the transition points in piecewise functions? 2. Why is there a closed circle at $$x = 2$$ and $$x = -2$$? 3. Can you describe the behavior of the graph as $$x$$ approaches -2 and 2? 4. What changes would occur if the intervals were strict inequalities (e.g., $$x < -2$$)? 5. How does the function behave for large values of $$x$$? **Tip:** In piecewise functions, always check for continuity at the transition points to ensure there are no gaps in the graph.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Functions
Graphing
Formulas
g(x) = 1 if x ≤ -2
g(x) = x if -2 < x ≤ 2
g(x) = 1 if x > 2
Theorems
Continuity of Piecewise Functions
Suitable Grade Level
Grades 9-12