What is the inverse of this equation?

[ g + 41 ] / 13 = √....====> [ g + 41 ]² / 13² = - ( x + 20) ====> x = - 20 - [ g + 41]² / 13²

for your initial domain you should have found x ≤ - 20

The problem does not require the inverse of the function. You need only its domain and range. The given function is injective, and its domain and range would be implicit, so it does have an inverse.

The implicit domain of g is the set of values on which the argument of the square root is non-negative.

-(x + 20) ≥ 0

x + 20 ≤ 0

x ≤ -20

Domain of g: (-∞, -20]

√[-(x + 20)] ≥ 0

13√[-(x + 20)] ≥ 0

13√[-(x + 20)] - 41 ≥ -41

g(x) ≥ -41

Range of g: [-41, ∞)

The domain and range of g⁻¹ are the range and domain of g, respectively.

Domain of g⁻¹: [-41, ∞)

Range of g⁻¹: (-∞, -20]

The domain of a function = the range of its inverse, and vice versa.  Since you can't square root a negative, the domain of g(x) is x ≤ -20 so the range of g inverse is ≤ -20

The range of g(x) is g(x) ≥ -41 

so the domain of g inverse is x ≥ -41

To get the inverse you can switch x and g(x) [call it y] then solve for x:

x = 13 √(-(y + 20)) - 41

x + 41 = 13 √(-(y + 20))

(x + 41) / 13 =  √(-(y + 20))

((x + 41) / 13) ² = -(y + 20)

-((x + 41) / 13) ² = y + 20

-((x + 41) / 13) ²  - 20 = y = g inverse

IMO anyway