Understanding Piecewise Functions: Key Points and Examples

Math Problem Statement

give the 2 points for each

Given the piecewise function:

\begin{cases} -1 & \text{for } -4 < x < 1 \\ x + 2 & \text{for } x < -5 \end{cases}$$ We will identify two points for each of the defined pieces of the function. ### For $$f(x) = -1$$ when $$-4 < x < 1$$: 1. **Point 1:** Let's choose $$x = -3$$: - $$f(-3) = -1$$ - Point: $$(-3, -1)$$ 2. **Point 2:** Let's choose $$x = 0$$: - $$f(0) = -1$$ - Point: $$(0, -1)$$ ### For $$f(x) = x + 2$$ when $$x < -5$$: 1. **Point 1:** Let's choose $$x = -6$$: - $$f(-6) = -6 + 2 = -4$$ - Point: $$(-6, -4)$$ 2. **Point 2:** Let's choose $$x = -7$$: - $$f(-7) = -7 + 2 = -5$$ - Point: $$(-7, -5)$$ So, the points corresponding to each piece of the function are: - For $$-4 < x < 1$$, the points are $$(-3, -1)$$ and $$(0, -1)$$. - For $$x < -5$$, the points are $$(-6, -4)$$ and $$(-7, -5)$$. Do you have any questions or need further details? Here are 5 related questions: 1. Would you like to see how these points fit on a graph? 2. Are there other intervals or pieces of the function you want to explore? 3. Should we find the intersection of these points with other functions? 4. Do you need to determine the behavior of the function as $$x$$ approaches certain values? 5. Would you like to see the domain and range of this function? **Tip:** When selecting points from piecewise functions, ensure they fall within the specified intervals to correctly represent the function's behavior.

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