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The domain of a function tells us what values the input, in this case, x, can take. It is essentially the complete set of possible x-values where the function is defined and operational. In our example, the function is piecewise, meaning it's defined by different expressions for different parts of the x-domain. This particular function allows every possible x value to fit into one of its pieces without leaving any gaps because three pieces are defined:

• For values where x < -3, the function is represented by the expression x + 9.
• Between -3 and 3 (inclusive), the function is given by -2x.
• For x > 3, it's defined as a constant, -6.

No value of x is left out, so the overall domain is all real numbers, or \((-\infty, \infty)\). Understanding the domain is crucial because it sets the stage for accurately sketching the graph, ensuring you know where each part of the function operates.

Graph sketching can initially seem challenging, but it becomes manageable by breaking it into smaller parts - especially handy for piecewise functions. Here's how you can sketch a piecewise function:

• **Identify Breakpoints:** These are values where the definition of the function changes. In our example, breakpoints occur at x = -3 and x = 3. These points signal shifts or new sections in the graph.
• **Plot Individual Sections:** Each section of the function needs to be drawn according to its expression:
  • For x < -3, sketch the line f(x) = x + 9 until you reach, but do not include, x = -3.
  • For -3 ≤ x ≤ 3, sketch the line f(x) = -2x, beginning at x = -3 and ending at x = 3, both points included.
  • For x > 3, sketch the horizontal line f(x) = -6 starting just beyond x = 3.
• **Check for Continuity and Endpoints:** At each breakpoint, ensure you understand how the graph changes. Some sections might not connect smoothly; they may show jumps or open circles, like at x = 3.

Combining these detailed steps ensures that you capture how the function behaves across its entire domain.

Piecewise functions are those defined by multiple sub-functions, each attached to a specific interval of the main variable, x. They are powerful because they allow a function to adapt or behave differently over different segments of its domain. This concept can be a bit tricky to understand initially. To break it down, consider:

• **Segments with Different Rules:** Each segment of the domain has its own rule or formula, like different paths on a map leading to several destinations depending on where you start.
• **Clear Intervals:** Conditions or intervals for each formula are clearly defined. It’s crucial to understand these intervals to properly analyze and graph the function.

For the function we are considering, each part:

• x < -3 uses x + 9
• -3 ≤ x ≤ 3 switches to -2x
• x > 3 relies on a constant value, -6.

Think of piecewise definitions as rules that direct traffic: they tell you precisely how to proceed based on the current value of x. Navigating these segments requires attention to detail, making sure each formula is applied in its rightful place.