Chapter 0.2, Problem 9E

(a)

To determine

To find: The domain and range of the given function.

Answer to Problem 9E

The domain is equal to $x\in R-\left\{2\right\}$ and the range is equal to $\left(-\infty ,0\right)\cup \left(0,\infty \right)$ .

Explanation of Solution

Given information:

The function is $y=\frac{1}{x-2}$ .

Calculation:

The given function is $y=\frac{1}{x-2}$ .

To find the domain: The denominator should not be equal to zero.

It can be written as below:

  $\begin{array}{c}x-2\ne 0\\ x\ne 2\end{array}$

So, the $y$ exists for all values of $x$ except $2$ .

Therefore, the domain will be

  $x\in R-\left\{2\right\}$

To find the range: Rewrite the equation of function and solve for $x$ .

  $\begin{array}{c}x-2=\frac{1}{y}\\ x=\frac{1}{y}+2\end{array}$

The denominator should not be equal to zero, so $y$ , must not be equal to zero.

It can be written as below:

  $y\ne 0$

So, the $x$ exists for all values of $y$ except $0$ .

Therefore, the range will be

  $R=\left(-\infty ,0\right)\cup \left(0,\infty \right)$

Hence, the domain is $x\in R-\left\{2\right\}$ and the range is $\left(-\infty ,0\right)\cup \left(0,\infty \right)$ .

(b)

To determine

To sketch: The graph of the function $y=\frac{1}{x-2}$ .

Explanation of Solution

Given information:

The function is $y=\frac{1}{x-2}$ .

Calculation:

The given function is $y=\frac{1}{x-2}$ .

The given function has a vertical asymptote at $x=2$ .

Now, sketch the graph of the function below: