Chapter 5.4, Problem 7E

To determine

Graph each function. Show any asymptotes as dashed lines.

Answer to Problem 7E

Horizontal asymptote = $\mathrm{y}=0$

Vertical asymptote = $\mathrm{x}=0$

Explanation of Solution

Given :

The function is :

  $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{{\mathrm{x}}^2}$

Calculation:

Make a table of different values of function:

x	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{{\mathrm{x}}^2}$
1	1
-1	1
$\mathrm{x}\to 0^{+}$	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{{\mathrm{x}}^2}\to +\mathrm{\infty }$
$\mathrm{x}\to 0^{-}$	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{{\mathrm{x}}^2}\to +\mathrm{\infty }$
$\mathrm{x}\to +\mathrm{\infty }$	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{{\mathrm{x}}^2}\to 0$
$\mathrm{x}\to -\mathrm{\infty }$	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{{\mathrm{x}}^2}\to 0$

Plot the points and we get the graph of the function.

  

An asymptote is a value of a function that you can get very near to, but you can never reach.

So, the function has two asymptotes:

Horizontal asymptote = $\mathrm{y}=0$

Vertical asymptote = $\mathrm{x}=0$