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Find the possible values of a*b. (Posted on 2024-01-26) Difficulty: 3 of 5
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.

See The Solution Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Solution solution Comment 2 of 2 |
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

  Posted by Larry on 2024-01-26 12:52:21
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