Question

Sea: f(x) = mx + n f(f(x)) = 4x + 9 Si m > 0, hallar “m + n”

Answer 1

¡Buenas!

$f(x) = mx+n \\ \\ f(f(x)) = 4x+9 \\ \\ f(mx+n) = 4x+9 \\ \\ \\ \\ \textbf{Hagamos:} \ \ \ x =1 \\ \\ f(m(1)+n) = 4(1)+9 \\ \\ f(m+n) = 13 \\ \\ \\ \\ \textbf{Hagamos:} \ \ \ x = 0 \\ \\ f(m(0)+n) = 4(0)+9 \\ \\ f(n) = 9$

$\textbf{Datos obtenidos: } \\ \\ \boxed{ f(m+n)=13 } \\ \\ \boxed{ f(n)=9} \\ \\ f(m+n) = m(m+n)+n= 13 \\ \\ f(n) = m(n)+n =9$

$m^{2} + mn +n = 13 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \boldsymbol{ ...\ (1)} \\ \\ mn +n = 9 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \boldsymbol{ ...\ (2)} \\ \\ \textbf{Reemplazamos (2) en (1)} \\ \\ m^{2} + \underbrace{mn+n} = 13 \\ \\ \textrm{}\ \ \ \ \ \ \ \ \ \ \ \ 9 \\ \\ m^{2} +9=13 \\ \\ m^{2} = 4 \\ \\ m_{1} = 2 \ \ \ \ \ \ m_{2} = -2 \\ \\ \textrm{El problema nos dice que}\ "m"\ \textrm{es mayor que cero.} \\ \\ \boxed{m = 2}$

$mn +n = 9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \boldsymbol{ ...\ (2)} \\ \\ \textbf{Sustituyendo "m" en (2)} \\ \\ 2n+n = 9 \\ \\ 3n=9 \\ \\ n=3 \\ \\ \boxed{n=3} \\ \\ \textrm{Nos piden:}\ m+n \\ \\ m+n \\ \\ 2+3 \\ \\ 5$

RESPUESTA

$\boxed{5}$