Given function:

$f(x) = |x+1| - 1$

Domain:

The function involves an absolute value, which is defined for all real numbers. Therefore, the domain of $f(x)$ is the set of all real numbers.

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

Range:

The absolute value function $|x+1|$ always produces non-negative values, i.e., $|x+1| \geq 0$. Subtracting 1 from this value shifts the entire output downward by 1.

• For $x = -1$, $|x+1| = 0$, so $f(x) = 0 - 1 = -1$.
• For any $x \neq -1$, the value of $|x+1|$ is positive, and thus $f(x) = |x+1| - 1 \geq -1$.

Therefore, the minimum value of $f(x)$ is $-1$, and since $|x+1|$ can grow infinitely large as $x$ moves away from $-1$, the function's values can increase indefinitely.

$\text{Range of } f(x): [-1, \infty)$

Summary:

• Domain: $(-\infty, \infty)$
• Range: $[-1, \infty)$

Would you like further details or clarifications?

Here are some related questions for you:

1. What happens to the function if we add or subtract a constant outside the absolute value?
2. How do transformations like shifting and stretching affect the graph of the absolute value function?
3. How would you find the domain and range of a similar function involving a square or cube?
4. What is the behavior of this function near $x = -1$?
5. How does the graph of $f(x) = |x+1| - 1$ compare to $f(x) = |x+1|$?

Tip: When dealing with absolute value functions, always remember that the output is always non-negative, and any vertical shifts are determined by adding or subtracting a constant.