To find the domain and range of the function $f(x) = -|2x| + 7$, let’s break it down step by step:

1. Domain:

The function involves an absolute value, but there are no restrictions like divisions by zero or square roots of negative numbers. Therefore, the domain is all real numbers, because for any value of $x$, $|2x|$ is defined.

$\text{Domain}: \ (-\infty, \infty)$

2. Range:

The function $f(x) = -|2x| + 7$ is a reflection of the absolute value function $|2x|$ across the $x$-axis (due to the negative sign) and then shifted up by 7.

• The minimum value of $|2x|$ is 0 (when $x = 0$).
• Thus, the maximum value of $f(x)$ occurs when $x = 0$, giving $f(0) = -|2(0)| + 7 = 7$.
• As $|2x|$ increases as $x$ moves away from 0, the function $-|2x|$ will decrease, so $f(x)$ will take values smaller than or equal to 7.

Therefore, the range is: $\text{Range}: (-\infty, 7]$

3. Graphing:

The graph of $f(x) = -|2x| + 7$ is a "V" shape, opening downward. The vertex of the graph is at $(0, 7)$, and since the slope of $-|2x|$ is steeper than the typical $-|x|$, the "V" will be narrower than the standard absolute value function.

The key points for graphing:

• At $x = 0$, $f(0) = 7$.
• For $x = 1$, $f(1) = -|2(1)| + 7 = -2 + 7 = 5$.
• For $x = -1$, $f(-1) = -|2(-1)| + 7 = -2 + 7 = 5$.
• As $x \to \infty$ or $x \to -\infty$, the function approaches $-\infty$.

If you'd like, I can generate a graph of this function for better visualization.

---

Would you like a detailed graph of this function or further clarifications?

Here are 5 questions related to this:

1. What happens to the graph if the absolute value coefficient changes?
2. How would the graph shift if the constant was different, say $f(x) = -|2x| + 10$?
3. How does the negative sign in front of the absolute value impact the graph?
4. How would the domain change if the function involved a square root?
5. How can we determine the range of a function without graphing?

Tip: When dealing with absolute value functions, always check for vertex points and symmetry in the graph!