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Question

If f(x) = 2x + 3 and g(x) = 3x + 5 then (f ^-1 - g ^-1)(5) equals...??

Answer 1

f^-1 means inverse of f
let y=f(x)=2x+3
x=(y-3)/2
change variables
y=(x-3)/2 .... here y is inverse of f
f^-1 = (x-3)/2

Similarly,
let again y=g(x)=3x+5
x=(y-5)/3
g^-1=(x-5)/3
f^-1-g^-1(x)=(x-3)/2 - (x-5)/3
=(3x-9-2x+10)/6
=(x+1)/6
(f ^-1 - g ^-1)(5) =(5+1)/6
=1

Answer 2

f^-1(5) = x for which f(x) = 5, i.e. 2x + 3 = 5 => x = 1.
Similarly g^-1(5) = 0, since g(0) = 5.
So (f^-1 - g^-1)(5) = f^-1(5) - g^-1(5) = 1 - 0 = 1.