First, let's find the inverse of the function g. An inverse function simply reverses the operations of the original function. So, for the function g(x) = 3x - 8, to find the inverse we'll switch the role of x and y, i.e., y = 3x - 8 becomes x = 3y - 8. Solving for y gives us the inverse function, g^-1(x) = (x + 8) / 3. Evaluating the inverse function for the values of x in the relation h, provides the following mapping:
g_inverse = {g^-1(-5): 1, g^-1(10): 6, g^-1(-17): -3, g^-1(-20): -4}
Next, to find the inverse of the relation h we do something quite similar. An inverse simply switches all ordered pairs in a relation or function. Therefore, we switch the place of the x and y values, so our pairs look like this:
h_inverse = {8: -4, 6: -3, -9: 1, -2: 6}
In conclusion, the inverse mapping of the function g and the relation h are as follows:
Inverse of function g(x) = 3x - 8:
g_inverse = {-5: 1, 10: 6, -17: -3, -20: -4}
Inverse of relation h = {(-4,8),(-3,6),(1,-9),(6,-2)}:
h_inverse = {-9: 1, -2: 6, 6: -3, 8: -4}