Question

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.

Answer 1

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.