Given that f(x)=x^2+x+1, f(-1)=1 and g(x) is the inverse of f(x), find g'(1)

Question

When solving for the inverse how do you determine if the square root of the inverse function (+/- sqrt(x-3/4)-1/2) will be positive or negative?

Answer 1

switch x and y to get

x=y^2+y+1

then solve for the new y

complete the square or use the quadratic formula

y^2 +y +1/4 = 1/4-1 +x

y+1/2=+/-sqr(x-3/4)

y =g(x)= -1/2 +/-sqr(x-3/4), with x> or = 3/4

g(x)= -1/2 +/-sqr(x-3/4) is the inverse relation. there is no inverse function

if you want an inverse function, use either + or -, but not both

g'(x)= +/-1/sqr(x-3/4)

g'(1) = +/-1/sqr(1/4)

g'(1)= +1/2 or - 1/2