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

1 Answer
Jul 20, 2018

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

Explanation:

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]}