Question

In Exercises 5-12, (a) identify the domain and range and (b) sketch the graph of the function. $$y=\frac{1}{x-2}$$

Short answer

The domain of the function \(y = \frac{1}{x - 2}\) is all real numbers except 2, \(x ≠ 2\), and the range is all real numbers except zero. The graph is a hyperbola with a vertical asymptote at \(x = 2\) and a horizontal asymptote at \(y = 0\).

Step-by-step solution

Identify the Domain

The domain of the function identifies all the possible x-values that the function can accept as input. For this function, the denominator cannot equal zero because division by zero is undefined. Therefore, to find the domain, solve the equation \(x - 2 ≠ 0\) which simplifies to \(x ≠ 2\).

Identify the Range

The range of the function represents all the possible output values or y-values of the function. Note that \(y = \frac{1}{x - 2}\) can take any real number value except for zero. So, the range of the function is all real numbers except for zero.

Sketch the Graph

Removing all \(x = 2\) (the vertical line) and \(y = 0\) (the horizontal line) from the xy-plane, the graph of this function will be a hyperbola. Also, as \(x\) approaches 2 from the left (e.g, 1.9, 1.99, etc.), \(y\) also approaches negative infinity; and as \(x\) approaches 2 from the right (e.g., 2.1, 2.01, etc.) \(\frac{1}{x - 2}\) approaches positive infinity. This behavior creates a vertical asymptote at \(x = 2\) and a horizontal asymptote at \(y = 0\).