Final answer:
To find f(x) using f(ax+b)=(2x+4)/(4x+3), replace ax+b with x, solve for y in x=ay+b, and substitute (x-b)/a into the function. Then, equate (x+3)/(2x+1) to the result and compare coefficients to obtain a=2 and b=-1, ensuring both forms of f(x) are equivalent.
Step-by-step explanation:
To find f(x) in terms of a and b, given f(ax+b) = (2x+4)/(4x+3), we need to replace ax+b with x in the equation for f. This means that we would solve for x in terms of a and b, and then substitute back into the original function f(ax+b).
Starting with the given function f(ax+b), we assume that x is the input we replaced with ax+b. So, to find f(x) we need to find the input that when plugged into a and b will yield x. This means:
x = ay + b
Solving for y, we get:
y = (x-b)/a
So, if we substitute in the value (x-b)/a for x in the original equation for f(ax+b), we would have:
f(x) = (2((x-b)/a)+4)/(4((x-b)/a)+3)
Now, we are given that f(x) = (x+3)/(2x+1). To find values of a and b, we equate this to the expression we just found for f(x):
(x+3)/(2x+1) = (2((x-b)/a)+4)/(4((x-b)/a)+3)
By comparing coefficients, we would find that a must be 2 and b must be -1 to have identical functions f(x).