Use graphing technology to find the domain of the function f, of, x, equals, minus, vertical bar, x, plus, 1, vertical bar, plus, 8, .*f*(*x*)=−∣*x*+1∣+8.

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.

Discover how to find the domain of the function f(x) = -|x + 1| + 8 through algebraic and graphical analysis. Understand the behavior of the absolute value function and linear transformations. Suitable for Grades 10-12 students and anyone interested in function domains.

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