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.