Question

Find the domain and sketch the graph of the function. What is its range? $$ f(x)=\left\\{\begin{array}{ll} -x-1 & \text { if } x<-1 \\ 0 & \text { if }-1 \leq x \leq 1 \\ x+1 & \text { if } x>1 \end{array}\right. $$

Short answer

The domain of the function is \(D = \{x \in \mathbb{R}\}\), which means all real numbers. The graph consists of 3 pieces: 1. A straight line with a slope of -1 and a y-intercept at -1 for \(x < -1\). 2. A horizontal line (y=0) between -1 and 1, inclusive. 3. A straight line with a slope of 1 and a y-intercept at 1 for \(x > 1\). The range of the function is \(R = \{y \in \mathbb{R} : y \neq 1\}\), which means all real numbers except 1.

Step-by-step solution

Identify the domain of the function

The domain of a function consists of all possible input values (x) for which the function is defined. In this case, we can see that f(x) is defined for all possible x values, as there are no restrictions or division by zero. Thus, the domain of the function is \(D = \{x \in \mathbb{R}\}\), which means all real numbers.

Sketch the graph of the function

To sketch the graph of the function, we need to draw each piece of the function on the coordinate plane within its respective domain: 1. \(f(x) = -x - 1\) if \(x < -1\) 2. \(f(x) = 0\) if \(-1 \le x \le 1\) 3. \(f(x) = x + 1\) if \(x > 1\) For the first piece (1), the equation represents a straight line with a slope of -1 and a y-intercept at -1. We will draw this line only for the values of x less than -1. For the second piece (2), the equation represents a horizontal line (y = 0) between -1 and 1, inclusive. We will draw this line segment between the points (-1,0) and (1,0), including the endpoints. For the third piece (3), the equation represents a straight line with a slope of 1 and a y-intercept at 1. We will draw this line only for the values of x greater than 1.

Find the range of the function

The range of a function consists of all possible output values (y) when the function is applied to its domain. By analyzing the graph from step 2, we can see that the function covers all y-values from \(-\infty\) to 1, and 0 to \(\infty\). The y-value of 1 is only attained by the third piece of the function, and the value of 0 is shared by both the second and third pieces of the function. Thus, the range of the function is \(R = \{y \in \mathbb{R} : y \neq 1\}\), which means all real numbers except 1.