To find the domain of the function $f(x) = -|x + 1| + 8$, we can analyze the function algebraically as well as graphically.

1. Algebraic Analysis

The function $f(x) = -|x + 1| + 8$ is composed of basic operations including an absolute value, a linear term inside the absolute value, and addition. The absolute value function $|x + 1|$ is defined for all real numbers $x$. Thus, there are no restrictions on $x$ for the expression $-|x + 1| + 8$ to be defined.

2. Graphical Analysis

To confirm this, let's graph the function $f(x) = -|x + 1| + 8$ and observe its behavior.

• The graph of $f(x) = -|x + 1| + 8$ will be a V-shaped graph flipped upside down (because of the negative sign) and shifted to the left by 1 unit (because of $x + 1$) and up by 8 units.

Conclusion

Since the absolute value function and linear operations are defined for all real numbers, the domain of $f(x)$ is all real numbers $x$. In interval notation, the domain is:

$\text{Domain of } f(x): \quad (-\infty, \infty)$

Graphical Confirmation

Using graphing technology or software (like Desmos, GeoGebra, or a graphing calculator), you can graph the function $f(x) = -|x + 1| + 8$. The graph will show that $f(x)$ is defined for all $x$, confirming that the domain is indeed all real numbers.

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Would you like more details or have any other questions?

Here are some related questions you might find useful:

1. How do you find the range of the function $f(x) = -|x + 1| + 8$?
2. How does the graph of $f(x) = -|x + 1| + 8$ compare to the graph of $f(x) = |x|$?
3. What is the vertex of the graph of $f(x) = -|x + 1| + 8$?
4. How would the graph change if the function were $f(x) = |x + 1| + 8$?
5. Can you express $f(x) = -|x + 1| + 8$ as a piecewise function?

Tip: When analyzing functions, always check for domain restrictions like divisions by zero, square roots of negative numbers, or logarithms of non-positive numbers.