Is g the inverse function of f?

Question

Let f : [−2,∞[→ R, f(x) = √ x + 2 and g : R → R, g(x) = x^2 − 2.

(a) Determine f ◦ g.

(b) Determine g ◦ f.

(c) Is g the inverse function of f? Why or why not?

Answer 1

if you mean f: (sqrx) +2 then its inverse is not g(x)

g: x^2 -2

f(g(x)) = sqr(x^2-1)+2

g(f(x)) = (sqrx +2)^2 -2= x + 2sqrx + 4-2 = x+2sqrx +2

they look similar but not identical for all values of x

to be inverses, they should both equal x

they aren't inverses

try a specific example, let x=1

f(1) = sqr(1)+2 = 1+2 = 3

g(3) = 3^2 -2 = 9-2 = 7

3 doesn't = 7

to find the inverse, switch x and y

y =x^2 -2 becomes

x =y^2 -2, then solve for the new y

y^2 = x+2

y = sqr(x+2)

BUT you likely meant f(x) = sqr(x+2) then g and f are inverses

f(g(x)) = (sqr(x+2)^2 -2 = x+2 -2 = x

g(f(x)) = [sqr(x+2)]^2 -2 = x+2-2 = x