suppose that f(x)=ax+b for all real numbers a and b.Given that f(f(f(x)))=8x=21, what is the value of a & b?

Answer 1

f(x) = ax + b
f(f(x)) = a(ax + b) + b = a^2 x + ab + b
f(f(f(x))) = a(a^2 x + ab + b) + b = a^3 x + b(a^2 + a + 1)
Since f(f(f(x))) = 8x, we must have: a = 2 and b = 0

Answer 2

There are a pair of the thank you to try this, however the least difficult is probable to think of how f acts once you compose it with some function. think we've g(x) = cx+d, and we compose with f to get f(g(x)). this might supply us f(g(x)) = (ac)x + (advert+b). So we multiply the x coefficient via a, and multiply the consistent via a and then upload b. So if we've been to compose f with itself two times (giving f(f(f(x))), the x coefficient would be a^3. provided that all of us understand that's 8, then a = 2. Now the consistent term may be (a^2 + a + a million)*b = 7*b, so b = 3. So the unique function replaced into f(x) = 2x+3, and a+b = 5.