# If f(x)= px+q and f(f(f(x)))=8x+21 and if p and q are real #'s, then p+q equals Options : A) 2 B) 3 C) 5 D) 7 E) 11

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Given `f(x)= px + q`

`` First, we will find `f(f(x)). `

`==gt f(f(x))= f(px+q) = p(px+q) +q = p^2 x+ pq+q`

`` Now we will find `f(f(f(x)))= f(p^2 x + pq+q) `

`==gt f(p^2x + pq+q)= p(p^2x+pq+q) +q = p^3 x + p^2q +pq+q `

`==gt f(f(f(x)))= p^3 x + p^2 q+pq +q= 8x+ 21`

`` Now we will compare sides.

`==gt (p^3) x + (p^2q + pq +q) = 8x + 21 `

`==gt p^3 = 8 ==gt p= 2 `

`==gt p^2q + pq + q = 21`

`` But `p= 2 `

`==gt 4q + 2q + q = 21 `

`==gt 7q = 21 `

`==gt q= 21/7= 3 `

`==gt p= 2 and q= 3 `

`==gt p+q = 2+3 = 5`

`` **Then the answer is c) 5**

Thanks a lot! :)