For the functions f(x) = 2x + 3 and g(x) = 6x + 2, which composition produces the greatest output?
a- niether
b- both
c- f(g(x))
d- g(f(x))
I think it is d
I agree. I assume you did some math to get that idea...
To determine which composition produces the greatest output, we need to find the values of f(g(x)) and g(f(x)).
1. Start by finding the composition f(g(x)):
- Substitute g(x) into f(x): f(g(x)) = 2(g(x)) + 3
- Replace g(x) with its expression: f(g(x)) = 2(6x + 2) + 3
- Simplify: f(g(x)) = 12x + 4 + 3
- Combine like terms: f(g(x)) = 12x + 7
2. Repeat the process for the composition g(f(x)):
- Substitute f(x) into g(x): g(f(x)) = 6(f(x)) + 2
- Replace f(x) with its expression: g(f(x)) = 6(2x + 3) + 2
- Simplify: g(f(x)) = 12x + 18 + 2
- Combine like terms: g(f(x)) = 12x + 20
Now, compare the expressions for f(g(x)) and g(f(x)). We can observe that the coefficient of x is the same (12) in both expressions. However, the constant terms differ. In f(g(x)), the constant term is 7, while in g(f(x)), the constant term is 20.
Since the constant term affects the output of the functions, the composition g(f(x)) produces the greatest output.
Therefore, the correct answer is d- g(f(x)).