Verify that the functions f and g, are inverses of each other by showing that

f(g(x))=x and g(f(x))=x.

f(x)=4-x^3
g(x)=^3�ã4-x

just plug in the values for f and g

f(g) = 4 - g^3 = 4 - cbrt(4-x)^3 = 4 - (4-x) = x

g(f) = cbrt(4-f) = cbrt(4 - (4-x^3)) = cbrt(4-4+x^3) = cbrt(x^3) = x

To verify that f and g are inverses of each other, we need to prove that their composition yields the identity function.

Let's start by evaluating f(g(x)):

1. Substitute g(x) into f(x):
f(g(x)) = 4 - (g(x))^3

2. Substitute the expression for g(x) into the equation:
f(g(x)) = 4 - (^3�ã4-x)^3

3. Simplify the expression inside the parentheses:
f(g(x)) = 4 - (4-x)^3

4. Expand the cube:
f(g(x)) = 4 - (64 - 48x + 12x^2 - x^3)

5. Simplify further:
f(g(x)) = 4 - 64 + 48x - 12x^2 + x^3

6. Combine like terms:
f(g(x)) = -60 + 48x - 12x^2 + x^3

Now, let's evaluate g(f(x)):

1. Substitute f(x) into g(x):
g(f(x)) = ^3�ã4 - f(x)

2. Substitute the expression for f(x) into the equation:
g(f(x)) = ^3�ã4 - (4 - x^3)

3. Simplify the expression inside the parentheses:
g(f(x)) = ^3�ã4 - 4 + x^3

4. Combine like terms:
g(f(x)) = ^3�ã-4 + x^3

To verify that f(g(x)) = x and g(f(x)) = x, we need to compare these expressions to the identity function, which is x.

Comparing f(g(x)) to x:
-60 + 48x - 12x^2 + x^3 = x

Comparing g(f(x)) to x:
^3�ã-4 + x^3 = x

If we solve these equations and find that x is the solution, then f and g are inverses of each other.

It's important to note that solving these equations requires finding the respective roots of the polynomials. This can be done using various methods such as factoring, the quadratic formula, or numerical approximations.