SOLUTION: Let f(x) and g(x) be functions. Find c if (f \circ g)(x) = (g \circ f)(x) for all x, where f(x) = 3x - 4 and g(x) = 5x + c.

Algebra ->  Functions -> SOLUTION: Let f(x) and g(x) be functions. Find c if (f \circ g)(x) = (g \circ f)(x) for all x, where f(x) = 3x - 4 and g(x) = 5x + c.      Log On


   

Question 1208803: Let f(x) and g(x) be functions. Find c if
(f \circ g)(x) = (g \circ f)(x)
for all x, where f(x) = 3x - 4 and g(x) = 5x + c.
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

is the same as writing f(g(x))
For more info, search out "function composition".

We're told that
f(g(x)) = g(f(x))
where
f(x) = 3x-4
g(x) = 5x+c

So,
f(x) = 3x-4
f(g(x)) = 3*g(x)-4
f(g(x)) = 3(5x+c)-4
f(g(x)) = 15x+3c-4
and
g(x) = 5x+c
g(f(x)) = 5*f(x)+c
g(f(x)) = 5(3x-4)+c
g(f(x)) = 15x-20+c

Equate those right hand sides and let's isolate c
15x+3c-4 = 15x-20+c
3c-4 = -20+c
3c-c = -20+4
2c = -16
c = -16/2
c = -8

This will mean g(x) = 5x+c updates to g(x) = 5x-8

As a check,
f(x) = 3x-4
f(g(x)) = 3*g(x)-4
f(g(x)) = 3(5x-8)-4
f(g(x)) = 15x-24-4
f(g(x)) = 15x-28
and,
g(x) = 5x-8
g(f(x)) = 5*f(x)-8
g(f(x)) = 5(3x-4)-8
g(f(x)) = 15x-20-8
g(f(x)) = 15x-28

We get 15x-28 each time, which proves that f(g(x)) = g(f(x)) is true for these functions.
It confirms we have the correct value of c.

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Answer: c = -8