What is the range of the function #f(x)= (x+7)/(2x-8)#?

1 Answer
Jul 14, 2017

Answer:

Undefined at #x=4#

#{x: -oo < x < oo," " x !=4}#

Explanation:

You are not 'allowed' to divide by 0. The proper name for this is that the function is 'undefined'. at that point.

Set #2x-8=0 => x=+4#

So the function is undefined at #x=4#. Sometimes this is referred to as a 'hole'.
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Domain and Range #-># letters d and r

In the alphabet d comes before r and you have to input (#x#) before you get an output (#y#).

So you consider the range as the values of the answer.

So we need to know the values of #y# as #x# tends to positive and negative infinity #->+oo and -oo#

As #x# becomes exceptionally big then the effect of the 7 in #x+7# is of no importance. Likewise the effect of -8 in #2x-8# becomes of no importance. My use of #-># means 'tends towards'

Thus as #x# tends towards positive infinity we have:
#lim_(x->+oo) (x+7)/(2x-8)->k=x/(2x)=1/2#

As #x# tends towards negative infinity we have:
#lim_(x->-oo) (x+7)/(2x-8)->-k=-x/(2x)=-1/2#

So the range is all values between negative infinity and positive infinity but excluding 4

In set notation we have:

#{x: -oo < x < oo," " x !=4}#