The function given in the image is:

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

Domain:

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Since there are no restrictions on the values of $x$ in the expression $|x| - 1$, the domain includes all real numbers.

Thus, the domain is: $\text{Domain} = (-\infty, \infty)$

Range:

The range of a function is the set of all possible output values (y-values).

The function $f(x) = |x| - 1$ is based on the absolute value function. The minimum value of $|x|$ is 0, which occurs when $x = 0$. Therefore, the minimum value of $f(x)$ is: $f(0) = |0| - 1 = 0 - 1 = -1$ As $|x|$ increases, the value of $f(x)$ will also increase without any upper bound. Thus, the range starts from $-1$ and goes to infinity.

So, the range is: $\text{Range} = [-1, \infty)$

Would you like further details or explanations?

Here are 5 related questions to deepen your understanding:

1. How does the range of the function $f(x) = |x| - 1$ compare to that of $f(x) = |x|$?
2. How would shifting the function up or down affect its range?
3. What is the effect of multiplying the function by a constant on its domain and range?
4. How does the graph of $f(x) = |x| - 1$ look in comparison to $f(x) = |x|$?
5. How would you define the domain and range if the function was $f(x) = |x - 1|$?

Tip: The absolute value function always outputs non-negative values, which is a key factor in determining its range.