Question

If fog(x) = 2x-1/x and g(x) = 5x + 2. Find a. f(x) b. The truth set of g inverse (x) + 3 = gof (x)​

Answer 1

Explanation:

Remember:

(fog)(x)= g(f(x))

a)

`(fog)(x)=2x-(1)/(x) =g(f(x))\\\\g(x)=5x+2\\\\Calculate\ g^(-1)(x)\\\\y=5x+2\\\\x=5y+2\ exchanging\ x\ and\ y\\\\y=(x-2)/(5) \\\\\boxed{g^(-1)(x)=(x-2)/(5)}\\\\`

`[(f\ o\ g)\ o\ g^(-1)](x)=[f\ o\ (g\ o \ g^(-1))](x)=f(x)\\\\g^(-1)(g(f(x))=g^(-1)(2x-(1)/(x) )=(2x-(1)/(x)-2)/(5) =(2x)/(5) -(1)/(5x)-(2)/(5) \\\\\\\boxed{f(x)=(2x)/(5) -(1)/(5x)-(2)/(5) }\\\\`

b)

`g^(-1)(x)+3=f(g(x))\\\\(x-2)/(5) +3=f(g(x))\\\\(x-2)/(5) +3=f(5x+2)\\\\(x-2)/(5) +3=(2(5x+2))/(5) -(1)/(5(5x+2))-(2)/(5) \\\\(x)/(5) -(2)/(5) +(15)/(5)=(10x)/(5) +(4)/(5) -(1)/(5(5x+2))-(2)/(5) \\\\9x-(1)/(5x+2)=11\\\\x\\eq -(2)/(5) \\\\`

`45x^2-37x-23=0\\\Delta=37^2+4*23*45=5509\\\\Sol=\{ (37-√(5509) )/(90) ,(37+√(5509) )/(90) \}`