f(x) = ax + b , f(ax + b) ) x +1 - what's a & b ?

Question

Let f(x) = ax + b

Pick numbers for a and b so that f(ax + b) = x + 1

 

 

Answer 1

Your title line is not a good representation of what the problem statement is, as given in the body of your submitted problem.

 

f(ax+b) = a[ax+b] + b=a²(x) + (ab+b)

 

If this right side must equal ax+b, then we equate coefficients of x and also we equate the constants, giving

a² = 1 & b(a+1) = 1.

 

If we choose a=1, then b must = ½.  (If a = -1 were chosen, then no value of b could satisfy the second equation).

Answer 2

If f(x) = ax + b, then f(ax + b) = a(ax + b) + b = a^2 x + ab + b. So we want a^2 = 1 and ab + b = 1. If a^2 = 1, then a = 1 or a = -1. Suppose a = 1. Then ab + b = 1 implies b + b = 1, or b = 1/2. So a = 1, b = 1/2 is a solution.

 

Note we actually cannot have a = -1, for then the equation ab + b = 1 would give -b + b = 1 or 0 = 1, which is impossible.