What is the range of #y=(-4x-3)/(x-2)#?

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Answer 1 · 2015-06-04T22:54:25

#f(x) = y = (-4x-3)/(x-2)#

#= ((-4x+8)-11)/(x-2)#

#= (-4(x-2)-11)/(x-2)#

#= -4-11/(x-2)#

#f(x)# can take any value except #-4# (to which #f(x)# is asymptotic as #x -> +- oo#).

So the range is #RR# \ #{-4}#

or in interval notation: #(-oo, -4) uu (-4, oo)#

graph{(-4x-3)/(x-2) [-39.76, 39.8, -19.92, 19.83]}

More explicitly, we can express #x# in terms of #y# as

#x = 2-11/(y+4)#

which has an obvious excluded value at #y=4#