How do you find the domain and range of #g(x)= sin(arccos(4x)) + cos(arcsin(5x))#?

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Answer 1 · 2018-07-20T16:41:37

Domain: #-1/5 <= x <= 1/5#. Range: #y in [ - 3/5, 0 ] U [ 3/5, 2] #
Please write your answer, separately.

Here, #-1 <= 4x <= 1 rArr -1/4 <= x <= 1/4# and, likewise,

#-1 <= 5x <= 1 rArr -1/5 <= x <= 1/5#. Combining,

#-1/5 <= x <= 1/5 rArr# both 4x and 5x in Q_1# or #Q_2#, wherein

sine is #+-# and cosine is non-negative..

#y = sin(arccos(4x))+cos(arcsin(5x))#

#= sin(arcsin(+-sqrt(1-16x^2))+ cos(arccos(sqrt(1-25x^2))#

#=+-sqrt( 1- 16x^2) + sqrt( 1- 25x^2 )#

Combined graph:
graph{(y-sqrt( 1- 16x^2) - sqrt( 1- 25x^2 ))(y+sqrt( 1- 16x^2) - sqrt( 1- 25x^2 ))=0[-0.5 0.5 -3 3]}