Use graphing technology to find the domain of the function f, of, x, equals, vertical bar, x, plus, 5, vertical bar, plus, 1, .*f*(*x*)=∣*x*+5∣+1.

To find the domain of the function \( f(x) = |x + 5| + 1 \), let's analyze the expression step-by-step. The domain of a function is the set of all possible input values (x-values) for which the function is defined.

### Analyzing the Function \( f(x) = |x + 5| + 1 \):

1. **Absolute Value Function**: The function involves an absolute value expression \( |x + 5| \).
   - The absolute value \( |x + 5| \) is defined for all real numbers \(x\).
   - Adding 1 to the absolute value does not change the domain, as addition is defined for all real numbers.

2. **Understanding the Operations**:
   - \( x + 5 \) is a linear expression and is defined for all real numbers.
   - The absolute value of any real number is also defined for all real numbers.

### Conclusion:

Since there are no restrictions on \(x\) for either the absolute value operation or the addition, the domain of the function \( f(x) = |x + 5| + 1 \) is **all real numbers**.

\[
\text{Domain} = (-\infty, \infty)
\]

### Using Graphing Technology:

If you were to use graphing technology (such as a graphing calculator or software like Desmos), you would see a V-shaped graph that starts from the point (-5, 1) and extends infinitely in both directions along the x-axis. The function is defined for every value of \(x\), confirming that the domain is indeed all real numbers.

Would you like to see more details or have any additional questions?

### Related Questions:

1. How would the domain change if the function were \( f(x) = \sqrt{|x + 5| + 1} \)?
2. What is the range of the function \( f(x) = |x + 5| + 1 \)?
3. How do absolute value functions typically behave graphically?
4. Can the domain of a function be affected by operations like division or taking a square root?
5. What are some common transformations of absolute value functions?

### Tip:

When working with absolute value functions, remember that the absolute value operation outputs only non-negative values, but it does not impose any restrictions on the domain unless combined with other operations like square roots or divisions.

Learn how to find the domain of the function f(x) = |x + 5| + 1 with a detailed step-by-step solution. Understand the behavior of absolute value functions and explore how graphing technology confirms the domain. Suitable for students in Grades 10-12.

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