Pré-Vestibular ⇒ (UFV) Função Composta

[robertosep]

Seja a função real [tex3]f[/tex3] tal que [tex3]f(x+2) = f(x)+\frac{5}{6}[/tex3] e [tex3]f(0)=\frac{5}{4}.[/tex3] Pode -se afirmar que [tex3]f(12)[/tex3] vale:

a) [tex3]\frac{77}{6}[/tex3]
b) [tex3]\frac{25}{4}[/tex3]
c) [tex3]\frac{65}{6}[/tex3]
d) [tex3]\frac{53}{4}[/tex3]
e) [tex3]\frac{19}{12}[/tex3]

[rodrigosnt]

• [tex3]f(12) = f(10+2) = f(10) + \frac{5}{6}\\
  f(12) = f(8+2) + \frac{5}{6} = f(8) + \frac{5}{6} + \frac{5}{6}\\
  f(12) = f(6+2) + \frac{5}{6} + \frac{5}{6} = f(6) + \frac{5}{6} + \frac{5}{6} + \frac{5}{6}\\
  f(12) = f(4+2) + \frac{5}{6} + \frac{5}{6} + \frac{5}{6} = f(4) + \frac{5}{6} + \frac{5}{6} + \frac{5}{6} + \frac{5}{6}\\
  f(12) = f(2+2) + \frac{5}{6} + \frac{5}{6} + \frac{5}{6} + \frac{5}{6} = f(2) + \frac{5}{6} + \frac{5}{6} + \frac{5}{6} + \frac{5}{6} + \frac{5}{6}\\
  f(12) = f(0+2) + \frac{5}{6} + \frac{5}{6} + \frac{5}{6} + \frac{5}{6} + \frac{5}{6} = f(0) + \frac{5}{6} + \frac{5}{6} + \frac{5}{6} + \frac{5}{6} + \frac{5}{6} + \frac{5}{6}\\
  f(12) = f(0) + 6\cdot\frac{5}{6}\\
  f(12) = \frac{5}{4} + 6\cdot\frac{5}{6} = \frac{5}{4} + 5\\
  f(12) = \frac{25}{4}[/tex3]

Resposta: (b).