plzz help, urgent (calculus)

Question

evaluate tan(sin^-1(1/4)).

g(x)+ (x-1)/(x+2).

a) what is the domain of g(x)

b) verify (with sketch and or other argument) that g is a one-to-one function.

c)find a formula for the inverse function. i.e., g^-1(x). d) find the range of g and g-1

Answer 1

tan(arcsin(1/4)) = (sqr15)/15 = about .2582

g(x)= (x-1)/(x+2) has domain x=all real numbers except -2, as it makes the denominator = 0

It's a rectangular hyperbola with vertical asymptote x=-2 and horizontal asymptote y=1

it's a one to one function, as it's impossible to intersect the graph with a vertical line that intersects the graph in more than one point

and it's impossible to intersect the graph with a horizontal line that intersects the graph in more than one point

it passes the vertical and horizontal line tests

inverse of g(x) is found by switching x and y, then solving for the new y

y= (x-1)/(x+2)

x = (y-1)/(y+2)

xy +2x = y-1

xy-y = -2x-1

y = -(2x+1)/(x-1) = g^-1(x) = the inverse function of g(x)

range of the inverse = domain of g(x) = all real numbers except x=-2

range of g(x) = domain of the inverse = all real numbers except x= 1

graphically the inverse of g(x) is a reflection of g(x) over the line y=x