Question

Find the domain and sketch the graph of the function. What is its range? $$ h(x)=\sqrt{x^{2}-1} $$

Short answer

The domain of the function \(h(x) = \sqrt{x^2 - 1}\) is \(x \le -1\) or \(x \ge 1\), and the range is \(h(x) \ge 0\). The graph consists of two separate U-shaped branches that are symmetric about the y-axis and non-negative.

Step-by-step solution

Identify the function and its characteristics

This is a square root function with an argument consisting of a quadratic expression. The square root function can only accept non-negative arguments (i.e., the argument must be greater than or equal to 0).

Determine the domain of the function

To find the domain, we need to find the values of x for which the argument of the square root function is non-negative. So, we need to solve the inequality for x as follows: \(x^2 - 1 \ge 0\) Factor the quadratic \((x - 1)(x + 1) \ge 0\) Now we can observe that the inequality holds when either both factors are greater than or equal to 0, or both factors are less than or equal to 0. Hence, the domain of the function is: \(x \le -1\) or \(x \ge 1\)

Find the range of the function

The range is the set of all possible output values for the function. Since the square root function is always non-negative and we have a quadratic expression within the square root, it's evident that the range must include all non-negative values. The minimum value attained by the function will be when the argument is equal to 0. Thus, the range of the function is: \(h(x) \ge 0\)

Sketch the graph

To sketch the graph of the function, we consider the following key points: 1. The domain of the function is \(x \le -1\) or \(x \ge 1\). So, there will be two separate portions of the graph. 2. The function is non-negative, so it cannot have any negative values on the y-axis. 3. The graph of \(y = h(x)\) will be symmetric about the y-axis since it's an even function. Based on these key points, the graph of the function can be sketched as: - On the left portion, the graph starts from x = -1, where h(x) = 0, and extends to the left (negative X-axis direction), without crossing the x-axis. - On the right portion, the graph starts from x = 1, where h(x) = 0, and extends to the right (positive X-axis direction), without crossing the x-axis. The resulting graph will look like two separate U-shape branches that are symmetric and non-negative in nature.