Question

let f(x)=-(2/3)x-2 and g(x)=ax+b. Find a and b so f(g(x))=x holds for all values of x

Answer 1

To find f(g(x)), substitute ax+b for x in f(x), and set this equal to x, giving:\[f(g(x))=(-2/3)(ax+b)-2=x\]Rearranging:\[(1+2a/3)x=-(2b/3+2)\]If this is to hold for all x, it must hold, for example, for x=0, therefore:\[2b/3+2=0\]giving b=-3. Putting this into the equation above, we have:\[(1+2a/3)x=0\]for all x. This can only be true if (1+2a/3)=0, so a=-1.5