Math help!!!!!!!!!!!?

Question

Suppose that f(x)=ax+b, where a and b are real numbers. Given that f(f(f(x)))=8x+21, what is the value of a+b?

Answer 1

ok. so f(f(f(x))), given that f(x) = ax + b means:

a(a(ax + b) + b) + b= 8x + 21

now that is:
a(a^2x + ab + b) + b = 8x + 21

and that is:

a^3x + a^2b + ab + b= 8x + 21

now, since a^3 is the coefficient of x and so is 8, this means:

a^3 = 8
which means a = 2

now:

a^2b + ab + b= 21

plug in 2 for a:

4b + 2b + b = 21

7b = 21

b=3

Answer 2

Suppose that f(x)=ax+b, where a and b are real numbers. Given that f(f(f(x)))=8x+21, what is the value of a+b

f(x) = ax + b
f(f(x) = a(ax + b) + b
= a²x + ab + b

f(f(f(x))) = a(a²x + ab + b) + b
= a³x + a²b + ab + b

BUT f(f(f(x)))=8x+21

So a³ = 8 and a²b + ab + b = 21

So a = 2 So b(a²+ a + 1) = 21 ie b*7 = 21 so b=3

So a + b = 5 and f(x) = 2x + 3

Check f(f(x)) = 2(2x + 3) + 3 = 4x + 9
Whence f(f(f(x))) = 2(4x + 9) + 3
= 8x + 21 Woooohooooooooooo!!!!!!!!

Answer 3

1) f(x) = ax + b

2) f(f(x)) = a(ax+b)+b

3) f(f(f(x))) = a(a(ax+b)+b)+b = 8x +21

4) a(a^2 * x + ab + b) + b = a^3 * x + a^2 * b + ab + b = 8x + 21

5) a^3 * x + b(a^2 + a + 1) = 8x + 21

6) (a^3 - 8)x +[ b( a^2 + a +1 ) - 21 ] = 0

6) since a and b are independent, 5) is only possible if
a^3 = 8
b(a^2 + a + 1) = 21

7) a = 2; b = 21 / ( 4 + 2 + 1 ) = 21 / 7 = 3

8) a + b = 5

Answer 4

so, ax + b replaces x.
a[a(ax + b) + b] + b = 8x + 21
a[a^2 x + ba + b] + b = 8x + 21
a^3 x + ba^2 + ba + b = 8x + 21
a^3 x + ba^2 + ba + b = 8x + 21
since the coefficient in front of x = a ^ 3 and 8. a = 2 since the cube root of 8 = 2.
ba^2 + ba + b = 21 since there is no variable next to these.
replace a with 2:
4b + 2b + b = 21
7b = 21
b = 3
So, a + b
=2+3
= 5.

Answer 5

charmed potato, if you don't know don't answer.

With that aside, the way this works is you plug it into itself. i.e. f(f(f(x))=a(a(ax+b)+b)+b

Simplifying, that=(a^3)x+((a^2)b+ab+b)=8x+21, meaning that a^3=8, meaning a=2. Also, it means that a^2b+ab+b=21. Substituting 2 in for a, we get 7b=21, meaning b=3.

Therefore, a+b=5.

Answer 6

f(f(x)) = a(ax+b)+b = aax+ab+b
f(f(f(x))) = a(aax+ab+b)+b = aaax+aab+ab+b = 8x + 21
so
aaax = 8x --> a = 2
so
aab+ab+b = 21 --> 4b+2b+b = 21 --> 7b = 21 --> b=3

Answer 7

f(f(f(x)))=8x+21
f(x)=ax+b

f(f(x)) = a(ax+b) +b = a^2x +ab + b
f(f(f(x))) = a^2(ax+b) + ab + b = a^3x +a^2b +ab + b = 8x + 21

==> a=2, b=3

Answer 8

i don't know---------sorry