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Mgr. Jozef Vozár 2007

Exponenciálna funkcia. Mgr. Jozef Vozár 2007. Definícia. Exponenciálnou funkciou budeme nazývať každú funkciu určenú vzťahom f: y = a x Kde a ε R + { 1 } . Grafom funkcie je exponenciála. Graf pre a > 1. a = 2. a = 3. a = 4. Graf pre 0 <a<1. a = 1/2. a = 1/3. a=1/4.

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Mgr. Jozef Vozár 2007

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  1. Exponenciálna funkcia Mgr. Jozef Vozár 2007

  2. Definícia Exponenciálnou funkciou budeme nazývať každú funkciu určenú vzťahom f: y = ax Kde a ε R+\{ 1 }. Grafom funkcie je exponenciála

  3. Graf pre a > 1 a = 2

  4. a = 3

  5. a = 4

  6. Graf pre 0 <a<1 a = 1/2

  7. a = 1/3

  8. a=1/4

  9. Vlastnosti D(f) = R H(f) = R+ Prechádzajú bodom [0;1] Pre a>1 sú rastúce Pre 0<a<1 sú klesajúce Exponenciálna fcia je prostá

  10. Exponenciálne rovnice Pri riešení využívame • To že exponenciálna funkcia je prostá ax = ay x = y 2. Vety o operáciách s mocninami

  11. Príklad č.1 3x + 2 = 32x + 7 x + 2 = 2x + 7 x = 5

  12. Príklad č.2 2 2x - 1 = 8 2 2x – 1 = 23 2x – 1 = 3 x = 2

  13. Príklad č.3 4x + 2x+1 = 80 t = 2x t2 + 2t - 80 = 0 t1 = 4 t2 = -10 2x = 4 2x = -10 x = 2

  14. Príklad č.4 4 x – 2 – 17.2 x – 4 + 1 = 0 2x = t t2/16 – 17.t/16 + 1 = 0 /.16 t2 - 17 . t + 16 = 0

  15. R4 t1 = 11 2x = 11 x = log11/log2 x = 3,45

  16. R4 t2 = 6 2x = 6 x = log6/log2 x = 2,58

  17. Príklad 5 7.3 x+1 – 5x+2 = 3x+4 – 5x+3 7.3 x+1 -3x+4 =5x+2 – 5x+3 21.3x – 81. 3x = 25 . 5x – 125. 5x 3x .60 = 5x .100 (3/5)x = 5/3 x = - 1

  18. Príklad 6 xx = x Logaritmujeme rovnicu x.log x = log x x.log x - log x = 0 log x.(x – 1) = 0 log x = 0 x – 1 = 0 x = 1

  19. Príklad 7 3.4log x – 25.2log x + 8 = 0

  20. R7 t = 2log x 3.t2 – 25.t + 8 = 0 t1 = 8 t2 = 1/3

  21. R7 2log x = 8 log x = 3 x = 1000

  22. R7 2log x = 1/3 log x. log 2 = log 1 – log 3 log x = - log3/log2 x = 10 - log3/log2 x = 10-1.585 = 0.0026

  23. Príklad 8 2.4x - 5.2x + 2.4x-1 = 0

  24. Riešenie t = 2x 2.t2 – 5t + 2.t2/4 = 0/ .2 4.t2 - 10t + t2 = 0 5 t2 - 10t = 0 t1 = 0, t2 = 2 x = 1

  25. Príklad 9 4x + 3x + 2 = 4x+3 - 3x + 2

  26. Riešenie 2. 3x + 2 = 4x+3 -4x 2. 3x + 2 = 4x (64 – 1) 18. 3x = 63. 4x (3/4)x = 63/18 = 21/6 = 7/2 x log3/4 = log 7/2 x = - 4,36

  27. Príklad 10 xx – x-x = 3( 1 + x-x)

  28. Riešenie xx = t t – 1/t = 3 (1 +1/t) t – 4/t – 3 = 0 t2 – 3t – 4 = 0 t1 = 4, t2 = -1 xx = 4 x = 2

  29. Príklad 11 log (x + 1) + log (x – 1) – log x = log (x + 2) (x + 1) (x – 1) = x. (x + 2) x2 – 1 = x2 + 2x x = - 0,5 Sk.log(0,5) + log (-1,5) – log(-0,5) = log(1,5) Rovnica nemá korene.

  30. Riešenie (x + 1) (x – 1) = x. (x + 2) x2 – 1 = x2 + 2x x = - 0,5 Sk.log(0,5) + log (-1,5) – log(-0,5) = log(1,5) Rovnica nemá korene.

  31. Príklad 12 log(4x + 6) – log(2x – 1) = 1

  32. Riešenie 4x + 6 –––––– = 10 2x – 1 4x + 6 = 20x – 10 x = 1 Sk. log10 – log 1 = 1 x = 1

  33. Príklad 13 1 xlog x – 1 = 10( 1 - ––- ) xlog x

  34. Riešenie t =xlog x t – 1 = 10( 1 – 1/t) t2– 11t + 10 = 0 t1 = 10, t2 = 1

  35. Riešenie xlog x = 10 log x . log x = 1 log x = 1, log x = - 1, x = 10, x = 1/10

  36. Riešenie xlog x = 1 log x . log x = 0 log x = 0 x = 1

  37. Príklad 14 4x – 3x-1/2 = 3x+1/2 - 22x-1

  38. Riešenie 4x + 22x-1 = 3x-1/2 + 3x+1/2 22x(1 + ½) = 3x(3-1/2 + 31/2) (4/3)x = 8/271/2 x = (log8-1/2log27)/(log4-log3) x = 0,0625

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