Domain: All real numbers except 1. Range: All real numbers except 2.
Practice makes perfect
We want to draw the asymptotes and the graph of the given function. To do so, we will start by considering some possible transformations.
Transformations of f(x)= 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
Horizontal Translations
Translation right h units, h>0
y=1/x- h
Translation left h units, h>0
y=1/x+ h
Vertical Stretch or Shrink
Vertical stretch, a>1
y=a/x
Vertical shrink, 0< a< 1
y=a/x
Reflection
In the x-axis
y=- 1/x
Note that if the graph of the function is translated, the asymptotes are also translated in the same distance and direction. Consider now the function.
y=- 2/x- 1+ 2
The given function is a combination of several transformations. A horizontal translation right 1 unit, a vertical stretch by a factor of 2, a reflection in the x-axis, and a vertical translation up 2 units. Let's apply these transformations one at a time. We will start by translating the parent function, f(x)= 1x, right 1 unit.
Now, let's apply a stretch of factor 2.
The second to last transformation is a reflection in the x-axis.
The last transformation is a vertical translation up 2 units.
Finally, let's see just the graph of the given function and its asymptotes.
We see above that the vertical asymtpote is the line x=1, and the equation of the horizontal asymptote is y=2. Using this information, we can state the domain and range of the function.
Domain:& All real numbers except 1
Range:& All real numbers except 2
