Question

In Exercises 5-12, (a) identify the domain and range and (b) sketch the graph of the function. $$y=1+\frac{1}{x}$$

Short answer

The domain of the function \(y = 1 + 1/x\) is all real numbers except 0, i.e., \((-∞ ,0) \cup (0, +∞)\). The range of the function is all real numbers greater than 0, i.e., \((0, +∞)\). The graph of the function will portray these facts visually.

Step-by-step solution

Identify the Domain

The domain of a function is the set of all possible input values. Given the function \(y = 1+1/x\), observe that \(1/x\) is undefined for \(x = 0\). Therefore, all real numbers except 0 are in the domain.

Identify the Range

The range of a function is the set of all possible output values. Inspecting the function, as \(x\) increases or decreases without bound, \(1/x\) approaches 0, and \(y\) approaches 1 accordingly. When \(x\) is positive, \(1/x\) is positive, which added to 1 yields a number greater than 1. Similarly, when \(x\) is negative, \(1/x\) is negative, which added to 1 yields a number less than 1 but greater than 0. Therefore, the range of the function is \((0, +\infty)\)

Sketch the Graph

To sketch the graph, plot some points for \(x\) and compute the corresponding \(y\)-values. For example, if \(x = -1\), \(y = 1 + 1/(-1) = 0\). Similarly, if \(x = 1\), \(y = 1 + 1/1 = 2\). Therefore, some points on the graph are (-1, 0) and (1, 2). Plot these and other points to create a more accurate sketch. Remember, the graph will not touch the y-axis as \(x\) can't be zero.