Koja je domena i opseg f (x) = {x ^ 2 - 81} / {x ^ 2 - 4x}?

Odgovor:

# D_f = RR- {0,4} (- oo, 0) uu (0,4) uu (4, + oo) #, Raspon = #F (D_f) = (- oo, (81-9sqrt65) / 8 uu (81 + 9sqrt65) / 8 + oo) #

Obrazloženje:

#F (x) = (x ^ 2-81) / (x ^ 2-4 *) *

Da bismo definirali ovu funkciju, trebamo # X ^ 2-4 *! = 0 #

Imamo # X ^ 2-4 * = 0 # #<=># #x (x-4), = 0 # #<=># # (X = 0, X = 4) #

Tako # D_f = RR- {0,4} (- oo, 0) uu (0,4) uu (4, + oo) #

Za #x## InD_f #, #F (x) = (x ^ 2-81) / (x ^ 2-4 *) * #=# # ((X-9), (x + 9)) / (x ^ 2-4 *) *

#f (x) = 0 <=> (x = 9, x = -9) #

• # (X ^ 2-81) / (x ^ 2-4 *) = y # #<=># # X ^ 2-81 = y (x ^ 2-4 *) #

# X ^ 2-81 = yx ^ 2-4xy #

• Dodavanje #COLOR (zeleno) (4yx) # na obje strane,

# X ^ 2-81 + 4yx-YX ^ 2 #

• oduzimanjem #COLOR (crveno) (YX ^ 2) * s obje strane

# X ^ 2-81 + 4yx-YX ^ 2 = 0 # #<=>#

# X ^ 2 (1-il) + 4xy-81 = 0 #

Ovo je kvadratna jednadžba za #x# tako

# A = 1-il #

# B = 4y #

# C = -81 #

Trebamo # D-b ^ 2-4 * a * c> = 0 # #<=>#

# 16y ^ 2-4 (1-il) + (- 81)> = 0 # #<=>#

# 16y ^ 2 + 324 (1-il)> = 0 # #<=>#

# 16y ^ 2-324y + 324> = 0 # #<=>#

# 4y ^ 2-81y + 81> = 0 #

#y_ (1,2) = (- b + -sqrt (b ^ 2-4ac)) / (2a) #

#=# # (81 + -sqrt (6561-1296)) / 8 #

#=# # (81 + -sqrt (5265)) / 8 #

#=# # (81 + -9sqrt65) / 8 #

# 4y ^ 2-81y + 81> = 0 # #<=># # (Y <(81-9sqrt65) / 8 # ili #Y> = (81 + 9sqrt65) / 8) #

tako, #F (x) <(81-9sqrt65) / 8 # ili #F (x)> = (81 + 9sqrt65) / 8 #

Što znači, #F (D_f) = (- oo, (81-9sqrt65) / 8 uu (81 + 9sqrt65) / 8 + oo) #