Question

Find the domain and sketch the graph of the function. What is its range? $$ f(x)=|x|-1 $$

Short answer

The domain of the function \(f(x)=|x|-1\) is all real numbers, or \((-\infty,+\infty)\). The range of the function is \([-1,+\infty)\). The graph is a V-shaped curve with its vertex at the point (0, -1) and is symmetric about the y-axis.

Step-by-step solution

Identify the Domain of the Function

The domain of a function is the set of all possible x-values for which the function is defined. The absolute value function, \(|x|\), is defined for all real numbers, so there are no restrictions on the x-values. Adding or subtracting a constant does not change the domain, so the domain for \(f(x) = |x| - 1\) is all real numbers. Domain: \(\{x \; | \; x \in \mathbb{R}\}\), which can also be written as \((-\infty,+\infty)\)

Observe the Properties of the Absolute Value Function

The absolute value function, \(|x|\), has the following properties: 1. It is non-negative, meaning \(|x| \ge 0\) for all \(x\) 2. It is symmetric about the y-axis, meaning \(|x| = |-x|\) These properties will help us analyze the behavior of the given function \(f(x) = |x| - 1\).

Find the Range of the Function

The range of a function is the set of all possible y-values. We need to find the minimum and maximum values of \(f(x)\), or the lowest and highest points of the graph. Since the minimum value of the absolute function \(|x|\) is 0 (at x = 0), the minimum value of \(f(x) = |x| - 1\) will be when \(|x|\) is at its minimum. So, at x = 0, we find that \(f(x) = |0| - 1 = 0 - 1 = -1\). The function has a minimum value of -1. There is no maximum value for \(|x|\), so there will be no maximum value for \(f(x) = |x| - 1\). Therefore, the function increases indefinitely as x moves away from 0, approaching infinity. With this information, we can conclude that the range of the function is \([ -1, +\infty)\).

Sketch the Graph of the Function

To sketch the graph of the function, use the following steps: 1. Start by drawing the graph of the absolute value function, \(|x|\). 2. Next, translating the graph down by 1 unit (since we are subtracting 1 from the function). 3. Finally, plot the graph of \(f(x) = |x| - 1\), which is a V-shaped graph with its vertex at the point (0, -1). The graph will be symmetric about the y-axis, have a vertex at the point (0, -1), and extend to infinity on both sides. By following these steps, you should be able to sketch the graph of the function \(f(x) = |x| - 1\).