Let us consider the functions f(x) and g(x) such that:
f(x)=ax+b and g(x)=bx+a.
Given that:
f(g(50))=g(f(50)), then find all possible values of a*b.
f(g(x)) = a[bx+a] + b = abx + a^2 + b
g(f(x)) = b[ax+b] + a = abx + b^2 + a
50ab + a^2 + b = 50ab + b^2 + a
a^2 + b = b^2 + a
(a^2 - b^2) - (a-b) = 0 which factors as
(a+b)(a-b) - 1(a-b) = 0
(a-b)(a+b-1) = 0
a can be any real number
case 1: b = a; their product is a^2
so, the requested quantity can be any real number >= 0
case 2: b = 1 - a; their product is a - a^2
This is a concave downward parabola (in 'a') with maximum value of 1/4 when a=1/2
so, the requested quantity can be any real number <= 1/4
infinite solutions; the set of all real numbers
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Posted by Larry
on 2024-01-26 12:52:21 |
solution
