randRangeNonZero( -3, 3 ) randRangeNonZero( -5, 5 ) expr([ "*", M, "x" ]) expr([ "*", B, "x" ]) varFraction( "x", M ) varFraction( "x", -M ) varFraction( M, "x" ) varFraction( B, M ) varFraction( M, B ) varFraction( "y", M ) subFraction( B_OVER_M ) subFraction( M_OVER_B ) addFraction( B_OVER_M ) function( x ) { return M * x + B; } function( x ) { return ( x - B ) / M; }

f(x) = M_X + B para todos los números reales.

¿Qué es f^{-1}(x), la inversa de f(x)?

graphInit({range:10,scale:20,tickStep:1,labelStep:2,axisArrows:"<->"}),style({stroke:BLUE,strokeWidth:2},function(){plot(F,[-10,10])})

X_OVER_M + MINUS_B_OVER_M

  • expr([ "+", M_X, -B ])
  • expr([ "+", M_X, B ])
  • expr([ "+", B_X, M ])
  • expr([ "+", M_OVER_X, B ])
  • expr([ "+", X_OVER_M, B ])
  • expr([ "+", X_OVER_M, -B ])
  • X_OVER_M + MINUS_M_OVER_B
  • X_OVER_M + PLUS_B_OVER_M
  • X_OVER_NEG_M + MINUS_B_OVER_M
  • X_OVER_NEG_M + PLUS_B_OVER_M

f(x) = M_X + B para todos los números reales.

Escribe una expresión para f^{-1}(x), la inversa de f(x).

graphInit({range:10,scale:20,tickStep:1,labelStep:2,axisArrows:"<->"}),style({stroke:BLUE,strokeWidth:2},function(){plot(F,[-10,10])})
x / M - B / M

y = f(x), así, resolviendo para x en terminos de y obtenemos x=f^{-1}(y)

f(x) = y = expr([ "+", M_X, B ])

expr([ "+", "y", -B ]) = M_X

Y_OVER_M + MINUS_B_OVER_M = x

x = Y_OVER_M + MINUS_B_OVER_M

Entonces sabemos: f^{-1}(y) = Y_OVER_M + MINUS_B_OVER_M

var pos=function(e){return e>=1?"below right":e>0?"below":e>-1?"above":"above right"},fPos=pos(M),fInvPos=pos(1/M);style({stroke:RED,strokeWidth:2},function(){plot(F_INV,[-10,10])}),-1===M||1===M&&0===B||(style({color:BLUE,strokeWidth:1},function(){label(labelPos(F),"f(x)",fPos)}),style({color:RED,strokeWidth:1},function(){label(labelPos(F_INV),"f^{-1}(x)",fInvPos)}),style({stroke:"#aaa",strokeWidth:2,strokeDasharray:"- "},function(){plot(function(e){return e},[-10,10])}))

Renombra y ax: f^{-1}(x) = X_OVER_M + MINUS_B_OVER_M

Observa que f^{-1}(x) es simplemente f(x) reflejada a través del la línea y = x.