What is the range of the following function in interval notation:  f(x)=-4|2x+6|-7

To find the range of the function \( f(x) = -4|2x + 6| - 7 \), we proceed step by step:

### Step 1: Analyze the structure of the function.
The function consists of a constant multiple of an absolute value function, with a vertical shift.

- The term \( |2x + 6| \) represents the absolute value of a linear expression. Since absolute values are always non-negative, we know that:
  \[
  |2x + 6| \geq 0
  \]
- When multiplied by -4, this changes the direction of the inequality:
  \[
  -4|2x + 6| \leq 0
  \]
- The smallest value that \( -4|2x + 6| \) can take is \( 0 \) when \( |2x + 6| = 0 \), which happens when \( x = -3 \).

### Step 2: Shift the function by subtracting 7.
- Now we subtract 7 from the expression:
  \[
  f(x) = -4|2x + 6| - 7
  \]
  This shifts the entire function downward by 7 units. Therefore:
  \[
  f(x) \leq -7
  \]
  The maximum value of the function occurs when \( |2x + 6| = 0 \), which happens at \( x = -3 \). At this point:
  \[
  f(-3) = -4(0) - 7 = -7
  \]

- As \( x \) moves away from \( -3 \), \( |2x + 6| \) increases, and thus \( f(x) \) becomes more negative. Hence, the function decreases without bound.

### Step 3: Determine the range.
- The function reaches its maximum value of \( -7 \) at \( x = -3 \), and it decreases toward \( -\infty \) as \( x \) moves further away from \( -3 \).

Thus, the range of the function in interval notation is:
\[
\boxed{(-\infty, -7]}
\]

Do you want more details or have any questions?

Here are 5 related questions:
1. How do we find the domain of absolute value functions?
2. What is the effect of multiplying an absolute value function by a negative number?
3. How does vertical translation affect the range of a function?
4. How would the range change if the constant in the function were positive instead of negative?
5. What happens to the graph of \( f(x) \) if the \( -7 \) were replaced with another constant?

**Tip:** When dealing with absolute value functions, always first analyze the behavior of the function inside the absolute value before considering transformations like scaling and translations.

What is the range of the following function in interval notation: f(x)=-4|2x+6|-7

Explore how to determine the range of the function f(x) = -4|2x + 6| - 7. This solution explains key concepts like absolute value, vertical shifts, and multiplying by negative numbers. Ideal for high school students in Grades 10-12.

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