G(x)=x^2+2x+5,

Question

find the inverse function? 

Answer 1

The function g(x) = x² + 2x + 5

Change g(x) to y.

y = x² + 2x + 5

Inter change x and y.

x = y² + 2y + 5

Now solve for y.

y² + 2y + (5 - x) = 0

Solve the above equation using quadratic formula.

Compare it to ax² + bx + c = 0

x = y, a = 1, b = 2, c = 5 - x

Roots are y = [- b ±√(b² - 4ac)]/2a

b² - 4ac = 4 - 4(1)(5 - x) = 4 - 20 + 4x = - 16 + 4x

y = [ - 2 ± √(- 16 + 4x)]/2

y = [ - 2 ± 2√(- 4 +x)]/2

y = - 1 ± √(- 4 + x)

Inverse is y = - 1 ± √(x - 4)

Change y to g^-1(x)

g^-1(x) = - 1 ± √(x - 4).