How do you find the domain and range of #y = 3(x-2)/x#?

1 Answer
Aug 21, 2015

Answer:

Domain: #(-oo, 0) uu (0, + oo)#
Range: #(-oo, 3) uu (3, + oo)#

Explanation:

Right from the start, you can say that the domain of the function will not include #x=0#, since that would make the denominator of the fraction equal to zero.

This means that the domain of the function will be #RR - {0}#, or #(-oo, 0) uu (0, + oo)#.

To find if the range of the function has any restrictions, calculate the inverse of #y# by solving for #x#, then switching #x# with #y#

#y = (3x - 6)/x#

#y * x = 3x - 6#

#x (y-3) = 6 implies x = 6/(y-3)#

The inverse function will thus be

#y = 6/(x-3)#

As you can see, this is not defined for #x=3#, which means that your original function cannot take the value #y=3#. The range of the function will thus be #RR- {3}#, or #(-oo, 3) uu (3, + oo)#.

graph{(3(x-2))/x [-10, 10, -5, 5]}