160k views
3 votes
Given f(ax+b) = (2x+4)/(4x+3), find f(x) in term of a and b. If f(x) = (x+3)/(2x+1), obtain values of a and b.

1 Answer

3 votes

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).

User Tom Bates
by
8.2k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.