Given that f(x) = √(ax + 1) , with x ≥ -1/a and a > 0 and g(x) = (x+1)/x, with x ≠ 0.
if ( f~¹ • g~¹) (3) = -⅜ ,
find : a²+ 3a -3 !​

Step-by-step explanation:

first thing I assume by f~¹ you meant however

we want to find a²+3x-3 for the given condition. with the composite function condition we can do so

Finding the inverse of f(x):

substitute y for f(x):

interchange:

square both sides:

cancel 1 from both sides:

divide both sides by a:

substitute f^-1 for y:

finding the inverse of g(x):

substitute y for g(x)

interchange:

cross multiplication

cancel 1 from both sides

factor out y:

divide both sides by 1-x:

substitute g^-1 for y:

remember that

therefore we obtain:

since (f~¹•g~¹)(3)=-⅜ thus substitute:

simplify parentheses:

simplify square:

simplify substraction:

simplify complex fraction:

get rid of - sign:

divide both sides by 3:

cross multiplication:

divide both sides by 4:

as we want to find a²+3a-3 substitute the got value of a:

simplify square:

simplify multiplication:

simplify addition:

simplify substraction:

and we are done!

ok

step-by-step explanation:

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step-by-step explanation: