Domain: All real numbers except x=0 Range: All real numbers except y=4
Practice makes perfect
We want to draw the asymptotes and the graph of the given function. We will start by considering some possible transformations.
Transformations of y= 1x, x≠ 0
Vertical Translations
Translation up k units, k>0
y=1/x+ k
Translation down k units, k>0
y=1/x- k
Vertical Stretch or Shrink
Vertical stretch, a>1
y=a/x
Vertical shrink, 0< a< 1
y=a/x
Note that if the graph of the function is translated, the asymptotes are also translated in the same distance and direction. Consider the function.
y=3/x+ 4
The given function is a combination of transformations.
Vertical stretch by a factor of 3
Vertical translation up 4 units
Let's apply these transformations one at a time. We will start by applying the parent function, y= 1x, by a factor of 3.
Now, let's apply a vertical translation 4 units up.
Finally, let's look at the graph of the given function and its asymptotes alone.
We can see that the vertical asymtpote is the line x=0, and the equation of the horizontal asymptote is y=4. Using this information, we can state the domain and range of the function.
Domain:& All real numbers except x=0
Range:& All real numbers except y=4
