Question

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

Answer 1

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]}