Let f(x)=2x+7 and
g(x)=3x+c. Find c if (f#g)(x)=(g#f)(x) for all x. Thank you.
Lots of problems posted, with no sign of effort on your part.
No ideas at all?
I would have said something like that but have been having fun :)
She has an imaginative teacher!
To find the value of c such that (f#g)(x) = (g#f)(x) for all x, we need to evaluate the composite functions:
(f#g)(x) = f(g(x))
(g#f)(x) = g(f(x))
Let's substitute the given functions into these composite functions:
(f#g)(x) = f(g(x)) = 2(g(x)) + 7 = 2(3x + c) + 7 = 6x + 2c + 7
(g#f)(x) = g(f(x)) = 3(f(x)) + c = 3(2x + 7) + c = 6x + 21 + c
Now, set them equal to each other and solve for c:
6x + 2c + 7 = 6x + 21 + c
Subtract 6x from both sides:
2c + 7 = 21 + c
Subtract c from both sides:
2c - c + 7 = 21
Combine like terms:
c + 7 = 21
Subtract 7 from both sides:
c = 21 - 7
c = 14
So, the value of c that makes (f#g)(x) = (g#f)(x) for all x is 14.