Exercise 11 Page 512

We are asked to graph the given function. First, we will identify the $x\text{-}$ and $y\text{-}$intercepts, as well as the asymptotes. After that, we will state the domain and the range.

Graph

To graph the function, we will make a table of positive and negative values. Note that $x$ cannot be zero because division by zero is not defined.

$x$	$\text{-}\frac{10}{x}$	$y=\text{-}\frac{10}{x}$
$0.5$	$\text{-}\frac{10}{0.5}$	$\text{-}20$
$1$	$\text{-}\frac{10}{1}$	$\text{-}10$
$2$	$\text{-}\frac{10}{2}$	$\text{-}5$
$5$	$\text{-}\frac{10}{5}$	$\text{-}2$
$10$	$\text{-}\frac{10}{10}$	$\text{-}1$
$\text{-}0.5$	$\text{-}\frac{10}{\text{-}0.5}$	$20$
$\text{-}1$	$\text{-}\frac{10}{\text{-}1}$	$10$
$\text{-}2$	$\text{-}\frac{10}{\text{-}2}$	$5$
$\text{-}5$	$\text{-}\frac{10}{\text{-}5}$	$2$
$\text{-}10$	$\text{-}\frac{10}{\text{-}15}$	$1$

Because $x$ cannot be zero the graph will not cross the $y\text{-}$axis. Therefore, we need two curves to connect the points.

Intercepts

Consider the given function. Since the denominator cannot be $0,$ $x$ cannot be $0.$ Therefore, there is no $y\text{-}$intercept. Moreover, the numerator is never $0,$ so $y$ is never $0.$ There is also no $x\text{-}$intercept. We can confirm this by looking at the graph.

We can see that the curve does not cross either axis.

Asymptotes

Let's consider the graph of the given function.

Notice how the $y\text{-}$values get closer to $0$ as the absolute values of $x$ get larger. This tells us that the $x\text{-}$axis is a $\text{horizontal}$ $\text{asymptote}.$ Furthermore, notice that the absolute values of $y$ get very large as $x$ approaches $0.$ This means that the $y\text{-}$axis is a $\text{vertical}$ $\text{asymptote}.$

Domain and Range

Consider the given function. Since the denominator cannot be $0,$ $x$ also cannot be $0.$ The domain of the function is all real numbers except $0.$ Moreover, the numerator is never $0,$ so $y$ is never $0.$ The range is all real numbers except $0.$