How do you find domain and range for #f(x) = (4x - 7) /( 6 - 5x)#?

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Answer 1 · 2018-07-02T09:07:26

The domain is #x in (-oo, 6/5) uu(6/5, +oo)#.
The range is #y in (-oo, -4/5) uu(-4/5, +oo)#

The denominator must be #!=0#

#=>#, #6-5x!=0#

#=>#, #x !=6/5#

The domain is #x in (-oo, 6/5) uu(6/5, +oo)#

To find the range, let

#y=(4x-7)/(6-5x)#

#=>#, #y(6-5x)=4x-7#

#=>#, #6y-5xy=4x-7#

#=>#, #4x+5xy=6y+7#

#=>#, #x(4+5y)=6y+7#

#=>#, #x=(6y+7)/(5y+4)#

The denominator must be #!=0#

#=>#, #5y+4!=0#

#=>#, #y !=-4/5#

The range is #y in (-oo, -4/5) uu(-4/5, +oo)#

graph{(4x-7)/(6-5x) [-11.25, 11.25, -5.625, 5.625]}