To find the domain of the function $f(x) = |x + 5|$ using graphing technology, let's first analyze the function both algebraically and graphically.

1. Algebraic Analysis

The function $f(x) = |x + 5|$ is an absolute value function, which is defined for all real numbers. There are no operations within the function that restrict the domain (like division by zero or taking the square root of a negative number). Therefore, the function is defined for all values of $x$.

2. Graphical Analysis

To confirm this, we can graph the function $f(x) = |x + 5|$.

• The graph of $f(x) = |x + 5|$ is a V-shaped graph, with the vertex located at the point $(-5, 0)$. The graph opens upwards.

Using graphing technology (such as Desmos, GeoGebra, or a graphing calculator), plot the function $f(x) = |x + 5|$. You will observe that the function is defined for all values of $x$ and does not have any breaks, holes, or asymptotes in the graph.

Conclusion

The domain of $f(x) = |x + 5|$ is all real numbers, as the function is defined for any input $x$.

In interval notation, the domain is:

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

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

Tip: The absolute value function $|x + a|$ always produces a V-shaped graph. The vertex of the graph is at the point $(-a, 0)$.