Question

How do you sketch the graph of `f(x)=piarcsin(4x)`?

Answer 1

See graphs and explanation.

Explanation:

`y = pi arcsin (4 x )`. Inversely, `x = (1/4) sin ( y/pi )`.

The y-axis is the axis of this wave.

The wave-period = #(2pi)/(1/pi) = 2(pi)^2 = 19.74, nearly.

Amplitude =1/4 `rArr abs x <= 0.25.`.

The range : `y in [-(pi)^2/2, (pi)^2/2 ] = [ - 4.935, 4.935 ]`.

Domain ; `x in [ -1/4, 1/4 ]`.

See the graph of `y = pi arcsin (4 x )`, within the above guarding

limits, attributed to the convention

`arcsin ( 4 x ) in [ - pi/2, pi/2 ]`.
graph{y-3.14 arcsin (4x ) = 0}

Now see the graph for `y = pi (sin )^( -1 )( 4x )`, using the

common-to-both inverse `x = (1/4) sin (y/pi )`. Here, the inverse is

wholesome and, by sliding the graph `uarr datt`, you can see the

pixel\march to `+- oo`. This is not possible in the other graph

above, for .`y = pi arcsin (4 x )`..

graph{x-0.25 sin( y / 3.14)=0[ -1 1 -40 40] }

I expect that the hand calculators would soon use the

piecewise-wholesome `(sin)^(-1 )` operator and display display

`(sin)^(-1)(sin (-120^o))` as `-120^o` and not `- 60^o`, ..

.