Function f and g are defined by f(x) = #sqrt((x^2-2x))# and g(x) = 3x +4. The composite function is undefined for x ∈ ]a;b[ . Find the value of a and b?

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Answer 1 · 2018-07-05T16:45:45

The values of #a=-4/3# and #b=-2/3#

The functions are

#f(x)=sqrt(x^2-2x)#

#g(x)=3x+4#

The composite function is

#f(g(x))=f(3x+4)#

#=sqrt((3x+4)^2-2(3x+4))#

#=sqrt(9x^2+24x+16-6x-8)#

#=sqrt(9x^2+18x+8)#

The domain is

#9x^2+18x+8>=0#

Therefore,

The solution to this quadratic equation is

#x=(-18+-sqrt(18^2-4*9*8))/(18)#

#=(-18+-sqrt(18^2-4*9*8))/(18)#

#=(-18+-sqrt36)/18#

#=(-18+-6)/18#

Therefore,

the composite function is undefined for

#(-24/18, -12/18)#

#=(-4/3, -2/3)#

graph{sqrt(9x^2+18x+8) [-9.227, 1.873, -2.24, 3.31]}