https://file.aiquickdraw.com/mathbot/file/17273124517427avmjnp0.jpg

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.

State the domain and range for the function f(x) = |x| - 1.

Understand how to find the domain and range of the function f(x) = |x| - 1. Learn the concepts of absolute value, domain, and range in algebra, with step-by-step explanation. Suitable for students in Grades 9-12.

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