Problem 34 Find the domain and sketch the g... [FREE SOLUTION]

Chapter 0: Problem 34

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

Short Answer

Expert verified

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

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91影视

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

Short Answer

Step by step solution

Identify the Domain of the Function

Observe the Properties of the Absolute Value Function

Find the Range of the Function

Sketch the Graph of the Function

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