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.