How would you solve this composite function?

Question

Given f(x)=3x+k and g(x)=x-4/3, for what value of k is f(g(x))=g(f(x))?

Answer 1

(f.g)(x)=f(g(x))

=f(x-4/3)

=3(x-4/3)+k

=3x-4+k

Do (g.f)(x) the same way (switch f and g).

Answer 2

I think that here g(x) = (x- 4)/3.

So f(g(x)) = x - 4 + k; g(f(x)) = (3x + k - 4)/3. We get equation: x - 4 + k = x + k/3 - 4/3; 2/3k = 8/3; k = 4

Answer 3

Great... now we gotta do the problems twice:

once for g(x) = x - (4/3) and once for g(x) = (x-4)/3

I) g(x) = x - 4/3

f(g(x)) = 3(x - 4/3) + k

g(f(x)) = 3x+k - 4/3

3(x-4/3)+k = 3x + k - 4/3

3x - 4 + k = 3x + k - 4/3

k-4 = k - 4/3

no solution...

so it must be the latter

=======================================================

f(x) = 3x+k g(x) = (x-4)/3

f(g(x)) = 3(x-4)/3 + k

= x-4 + k

g(f(x)) = (3x+k-4)/3

x-4+k = (3x+k-4)/3

3x-12+ 3k = 3x+k - 4

3k-12 = k -4

2k-12 = -4

2k = 8

k=4