What value does a have to be for them to be inverses?

Question

h(x)=√x^3-2 g(x)=x^3+a

Answer 1

h(x) = sqr(x^3) -2

g(x) = x^3 +a

the inverse of h(x) is found by switching x and y and solving for the new y

y=sqr(x^3)-2

x =sqr(y^3)-2

sqr(y^3) = x+2

y^3 = (x+2)^2

y = (x+2)^(2/3) = x^3+a

a= (x+2)^(2/3)-x^3

g(x) = a +x^3

g(x) = (x+2)^(2/3) is the inverse function of h(x)=sqr(x^3)-2

check the answer with a point such as (x,y) = (0, 4^(1/3))

g(0) = 4^(1/3)

h(4^(1/3))= sqr(4)-2 =2-2= 0

h(g(x)) = x

but a=2, if the problem was reallly h(x) = cube root of (x -2)

maybe the problem has a typo or was mis-copied

generally use of letters a,b or c suggest a constant, not a variable

then g(x) = x^3 +2

check with a point such as (x,y) = (2,10)

h(g(2)) = h(10) = (10-2)^(1/3 =2

h(g(x)) = x

g(h(x) = x

g(x) cubes x then adds 2

h(x) subtracts 2 then takes a cube root

g and h are inverse operation, inverse functions