https://file.aiquickdraw.com/mathbot/file/17267042445746ivj1vv8.jpg

The given image defines a piecewise function \( g(x) \), where different formulas are applied depending on the value of \( x \). The function is structured as follows:

\[
g(x) =
\begin{cases}
3x + 54 & \text{for } -15 \leq x \leq -12 \\
18 & \text{for } -12 \leq x \leq -6 \\
x + 12 & \text{for } -6 \leq x \leq 6
\end{cases}
\]

### Explanation:

1. **First interval:**
   \[
   g(x) = 3x + 54 \quad \text{for } -15 \leq x \leq -12
   \]
   Here, the function is a linear equation. You plug in values of \( x \) between -15 and -12 into this equation to find the corresponding value of \( g(x) \).

2. **Second interval:**
   \[
   g(x) = 18 \quad \text{for } -12 \leq x \leq -6
   \]
   In this interval, the function is constant at \( g(x) = 18 \), regardless of the value of \( x \).

3. **Third interval:**
   \[
   g(x) = x + 12 \quad \text{for } -6 \leq x \leq 6
   \]
   In this range, the function is a linear equation with a slope of 1, meaning it increases by 1 unit for every 1 unit increase in \( x \).

Let me know if you would like further details on how to evaluate this function for specific values of \( x \).

### Related Questions:
1. How do you evaluate the piecewise function at specific points, like \( x = -13 \) or \( x = 0 \)?
2. What is the graph of this piecewise function like over the given intervals?
3. How do the slopes of the linear parts affect the behavior of the function?
4. What is the domain and range of this piecewise function?
5. How would you solve for \( g(x) = 0 \) in each interval?

**Tip:** When working with piecewise functions, always check which interval the value of \( x \) belongs to before evaluating the function.

The piecewise function g(x) with different formulas based on the value of x.

Explore how to work with a piecewise function defined by different equations for specific intervals of x. The function g(x) includes a linear equation, a constant value, and another linear equation. Learn how to evaluate it and its behavior across the intervals -15 ≤ x ≤ 6.

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