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.
g(50)=50*b + a
f(g(50))=a*(50*b+a) + b
f(50)=a*50 + b
g(f(50)) = b*(a*50+b) + a
a*(50*b+a) + b = b*(a*50+b) + a
50*a*b + a^2 + b = 50*a*b + b^2 + a
a^2 + b = b^2 + a
b^2 - b + a - a^2 = 0
b = (1 +/- sqrt(1 + 4*(a^2 - a)))/2
clearvars,clc
for a=-10:.1:10
b1 = (1 + sqrt(1 + 4*(a^2 - a)))/2;
b2 = (1 - sqrt(1 + 4*(a^2 - a)))/2;
fprintf('%5.1f %15.13f %15.13f\n',a,b1,b2)
end
by the table below shows
b = a or 1 - a
The product of a and b could be any real value.
a b1 b2
-10.0 11.0000000000000 -10.0000000000000
-9.9 10.9000000000000 -9.9000000000000
-9.8 10.8000000000000 -9.8000000000000
-9.7 10.7000000000000 -9.7000000000000
-9.6 10.6000000000000 -9.6000000000000
-9.5 10.5000000000000 -9.5000000000000
-9.4 10.4000000000000 -9.4000000000000
-9.3 10.3000000000000 -9.3000000000000
-9.2 10.2000000000000 -9.2000000000000
-9.1 10.1000000000000 -9.1000000000000
-9.0 10.0000000000000 -9.0000000000000
-8.9 9.9000000000000 -8.9000000000000
-8.8 9.8000000000000 -8.8000000000000
-8.7 9.7000000000000 -8.7000000000000
-8.6 9.6000000000000 -8.6000000000000
-8.5 9.5000000000000 -8.5000000000000
-8.4 9.4000000000000 -8.4000000000000
-8.3 9.3000000000000 -8.3000000000000
-8.2 9.2000000000000 -8.2000000000000
-8.1 9.1000000000000 -8.1000000000000
-8.0 9.0000000000000 -8.0000000000000
-7.9 8.9000000000000 -7.9000000000000
-7.8 8.8000000000000 -7.8000000000000
-7.7 8.7000000000000 -7.7000000000000
-7.6 8.6000000000000 -7.6000000000000
-7.5 8.5000000000000 -7.5000000000000
-7.4 8.4000000000000 -7.4000000000000
-7.3 8.3000000000000 -7.3000000000000
-7.2 8.2000000000000 -7.2000000000000
-7.1 8.1000000000000 -7.1000000000000
-7.0 8.0000000000000 -7.0000000000000
-6.9 7.9000000000000 -6.9000000000000
-6.8 7.8000000000000 -6.8000000000000
-6.7 7.7000000000000 -6.7000000000000
-6.6 7.6000000000000 -6.6000000000000
-6.5 7.5000000000000 -6.5000000000000
-6.4 7.4000000000000 -6.4000000000000
-6.3 7.3000000000000 -6.3000000000000
-6.2 7.2000000000000 -6.2000000000000
-6.1 7.1000000000000 -6.1000000000000
-6.0 7.0000000000000 -6.0000000000000
-5.9 6.9000000000000 -5.9000000000000
-5.8 6.8000000000000 -5.8000000000000
-5.7 6.7000000000000 -5.7000000000000
-5.6 6.6000000000000 -5.6000000000000
-5.5 6.5000000000000 -5.5000000000000
-5.4 6.4000000000000 -5.4000000000000
-5.3 6.3000000000000 -5.3000000000000
-5.2 6.2000000000000 -5.2000000000000
-5.1 6.1000000000000 -5.1000000000000
-5.0 6.0000000000000 -5.0000000000000
-4.9 5.9000000000000 -4.9000000000000
-4.8 5.8000000000000 -4.8000000000000
-4.7 5.7000000000000 -4.7000000000000
-4.6 5.6000000000000 -4.6000000000000
-4.5 5.5000000000000 -4.5000000000000
-4.4 5.4000000000000 -4.4000000000000
-4.3 5.3000000000000 -4.3000000000000
-4.2 5.2000000000000 -4.2000000000000
-4.1 5.1000000000000 -4.1000000000000
-4.0 5.0000000000000 -4.0000000000000
-3.9 4.9000000000000 -3.9000000000000
-3.8 4.8000000000000 -3.8000000000000
-3.7 4.7000000000000 -3.7000000000000
-3.6 4.6000000000000 -3.6000000000000
-3.5 4.5000000000000 -3.5000000000000
-3.4 4.4000000000000 -3.4000000000000
-3.3 4.3000000000000 -3.3000000000000
-3.2 4.2000000000000 -3.2000000000000
-3.1 4.1000000000000 -3.1000000000000
-3.0 4.0000000000000 -3.0000000000000
-2.9 3.9000000000000 -2.9000000000000
-2.8 3.8000000000000 -2.8000000000000
-2.7 3.7000000000000 -2.7000000000000
-2.6 3.6000000000000 -2.6000000000000
-2.5 3.5000000000000 -2.5000000000000
-2.4 3.4000000000000 -2.4000000000000
-2.3 3.3000000000000 -2.3000000000000
-2.2 3.2000000000000 -2.2000000000000
-2.1 3.1000000000000 -2.1000000000000
-2.0 3.0000000000000 -2.0000000000000
-1.9 2.9000000000000 -1.9000000000000
-1.8 2.8000000000000 -1.8000000000000
-1.7 2.7000000000000 -1.7000000000000
-1.6 2.6000000000000 -1.6000000000000
-1.5 2.5000000000000 -1.5000000000000
-1.4 2.4000000000000 -1.4000000000000
-1.3 2.3000000000000 -1.3000000000000
-1.2 2.2000000000000 -1.2000000000000
-1.1 2.1000000000000 -1.1000000000000
-1.0 2.0000000000000 -1.0000000000000
-0.9 1.9000000000000 -0.9000000000000
-0.8 1.8000000000000 -0.8000000000000
-0.7 1.7000000000000 -0.7000000000000
-0.6 1.6000000000000 -0.6000000000000
-0.5 1.5000000000000 -0.5000000000000
-0.4 1.4000000000000 -0.4000000000000
-0.3 1.3000000000000 -0.3000000000000
-0.2 1.2000000000000 -0.2000000000000
-0.1 1.1000000000000 -0.1000000000000
0.0 1.0000000000000 0.0000000000000
0.1 0.9000000000000 0.1000000000000
0.2 0.8000000000000 0.2000000000000
0.3 0.7000000000000 0.3000000000000
0.4 0.6000000000000 0.4000000000000
0.5 0.5000000000000 0.5000000000000
0.6 0.6000000000000 0.4000000000000
0.7 0.7000000000000 0.3000000000000
0.8 0.8000000000000 0.2000000000000
0.9 0.9000000000000 0.1000000000000
1.0 1.0000000000000 0.0000000000000
1.1 1.1000000000000 -0.1000000000000
1.2 1.2000000000000 -0.2000000000000
1.3 1.3000000000000 -0.3000000000000
1.4 1.4000000000000 -0.4000000000000
1.5 1.5000000000000 -0.5000000000000
1.6 1.6000000000000 -0.6000000000000
1.7 1.7000000000000 -0.7000000000000
1.8 1.8000000000000 -0.8000000000000
1.9 1.9000000000000 -0.9000000000000
2.0 2.0000000000000 -1.0000000000000
2.1 2.1000000000000 -1.1000000000000
2.2 2.2000000000000 -1.2000000000000
2.3 2.3000000000000 -1.3000000000000
2.4 2.4000000000000 -1.4000000000000
2.5 2.5000000000000 -1.5000000000000
2.6 2.6000000000000 -1.6000000000000
2.7 2.7000000000000 -1.7000000000000
2.8 2.8000000000000 -1.8000000000000
2.9 2.9000000000000 -1.9000000000000
3.0 3.0000000000000 -2.0000000000000
3.1 3.1000000000000 -2.1000000000000
3.2 3.2000000000000 -2.2000000000000
3.3 3.3000000000000 -2.3000000000000
3.4 3.4000000000000 -2.4000000000000
3.5 3.5000000000000 -2.5000000000000
3.6 3.6000000000000 -2.6000000000000
3.7 3.7000000000000 -2.7000000000000
3.8 3.8000000000000 -2.8000000000000
3.9 3.9000000000000 -2.9000000000000
4.0 4.0000000000000 -3.0000000000000
4.1 4.1000000000000 -3.1000000000000
4.2 4.2000000000000 -3.2000000000000
4.3 4.3000000000000 -3.3000000000000
4.4 4.4000000000000 -3.4000000000000
4.5 4.5000000000000 -3.5000000000000
4.6 4.6000000000000 -3.6000000000000
4.7 4.7000000000000 -3.7000000000000
4.8 4.8000000000000 -3.8000000000000
4.9 4.9000000000000 -3.9000000000000
5.0 5.0000000000000 -4.0000000000000
5.1 5.1000000000000 -4.1000000000000
5.2 5.2000000000000 -4.2000000000000
5.3 5.3000000000000 -4.3000000000000
5.4 5.4000000000000 -4.4000000000000
5.5 5.5000000000000 -4.5000000000000
5.6 5.6000000000000 -4.6000000000000
5.7 5.7000000000000 -4.7000000000000
5.8 5.8000000000000 -4.8000000000000
5.9 5.9000000000000 -4.9000000000000
6.0 6.0000000000000 -5.0000000000000
6.1 6.1000000000000 -5.1000000000000
6.2 6.2000000000000 -5.2000000000000
6.3 6.3000000000000 -5.3000000000000
6.4 6.4000000000000 -5.4000000000000
6.5 6.5000000000000 -5.5000000000000
6.6 6.6000000000000 -5.6000000000000
6.7 6.7000000000000 -5.7000000000000
6.8 6.8000000000000 -5.8000000000000
6.9 6.9000000000000 -5.9000000000000
7.0 7.0000000000000 -6.0000000000000
7.1 7.1000000000000 -6.1000000000000
7.2 7.2000000000000 -6.2000000000000
7.3 7.3000000000000 -6.3000000000000
7.4 7.4000000000000 -6.4000000000000
7.5 7.5000000000000 -6.5000000000000
7.6 7.6000000000000 -6.6000000000000
7.7 7.7000000000000 -6.7000000000000
7.8 7.8000000000000 -6.8000000000000
7.9 7.9000000000000 -6.9000000000000
8.0 8.0000000000000 -7.0000000000000
8.1 8.1000000000000 -7.1000000000000
8.2 8.2000000000000 -7.2000000000000
8.3 8.3000000000000 -7.3000000000000
8.4 8.4000000000000 -7.4000000000000
8.5 8.5000000000000 -7.5000000000000
8.6 8.6000000000000 -7.6000000000000
8.7 8.7000000000000 -7.7000000000000
8.8 8.8000000000000 -7.8000000000000
8.9 8.9000000000000 -7.9000000000000
9.0 9.0000000000000 -8.0000000000000
9.1 9.1000000000000 -8.1000000000000
9.2 9.2000000000000 -8.2000000000000
9.3 9.3000000000000 -8.3000000000000
9.4 9.4000000000000 -8.4000000000000
9.5 9.5000000000000 -8.5000000000000
9.6 9.6000000000000 -8.6000000000000
9.7 9.7000000000000 -8.7000000000000
9.8 9.8000000000000 -8.8000000000000
9.9 9.9000000000000 -8.9000000000000
10.0 10.0000000000000 -9.0000000000000
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Posted by Charlie
on 2024-01-26 11:43:54 |
possible solution
