Final answer:
To find g(x) given that (g \circ f)(x) = 5x + 1 and f(x) = 4x - 5, one can substitute f(x) into the composition, solve the resulting equation for g(u) where u = 4x - 5, and then replace u with x in the final expression to get g(x) = (5/4)x + (29/4).
Step-by-step explanation:
The given functions are f(x) = 4x - 5 and the composition of two functions (g \circ f)(x) = 5x + 1, which means g(f(x)).
To find g(x), we need to solve for g in the given composition. First, we substitute f(x) into (g \circ f)(x) and equate it to 5x + 1:
(g \circ f)(x) = g(f(x)) = g(4x - 5)
Given that (g \circ f)(x) = 5x + 1, we now have:
g(4x - 5) = 5x + 1
To find g(x), let u = 4x - 5. This implies x = (u + 5) / 4. Now, we can find g(u) knowing that g(4x - 5) = 5((u + 5)/4) + 1. After simplifying, we get:
g(u) = (5/4)u + (25/4) + 1
Since u is a stand-in for x in the function g, replacing u back with x gives:
g(x) = (5/4)x + (25/4) + 1 = (5/4)x + (29/4)