For the function f(x)=2x-6 and g(x)=5x+1, which composition produces the greatest output? a. both composition produce the same output. b. neither composition produces. c. f(g(x)) produces the greatest output. d. g(f(x)) produces the greatest output.

You have to model both options.

1) The composition of function is applying one function to the result of the other.

2) f(x) composed with g(x) is (f ° g) (x) = f [ g(x) ] means apply the function f to the result of g(x).

3) g(x) composed with f(x) is (g ° f) (x) = g [ f(x) ] means apply the function g to the result of f(x).

4) Do both: 

i) f (g(x) = 2 [ g(x) ] - 6 = 2 [ 5x + 1] - 6 = 10x + 2 - 6 = 10x - 4

ii) g (f(x) = 5 [ 2 (2x - 6) ] + 1 = 5 [ 4x - 12] + 1 = 20x - 60

5) Compare both outputs:

Is  20x - 60 > 10x - 4?

Solve the inequality:

20x - 10x > 60 - 4 ⇒ 10x > 56 ⇒ x > 5.6

Then:
i) for x = 5.6 both outputs are equal
ii) for x > 5.6 g(f(x) is greater
iii) for x < 5.6 f(x) is greater.