Use composition to show that the given functions are inverses (You must show both compositions). Help?

g(x)=x^2+3/2

To show that two functions are inverses, we need to demonstrate that when we compose them, we get the identity function.

Let's denote the given function as g(x) = x^2 + 3/2.

The first step is to find the inverse function of g(x). To do this, we will swap x with y and solve for y. This will give us the inverse function, denoted as g^(-1)(x):

x = y^2 + 3/2
x - 3/2 = y^2
√(x - 3/2) = y
g^(-1)(x) = √(x - 3/2)

Now, let's compute the composition of g(g^(-1)(x)):

g(g^(-1)(x)) = g(√(x - 3/2))
= (√(x - 3/2))^2 + 3/2
= x - 3/2 + 3/2
= x

The result of the composition g(g^(-1)(x)) is equal to x, which is the definition of the identity function. This shows that g(x) and g^(-1)(x) are indeed inverses.

Next, let's compute the composition of g^(-1)(g(x)):

g^(-1)(g(x)) = g^(-1)(x^2 + 3/2)
= √(x^2 + 3/2 - 3/2)
= √(x^2)
= |x|

The result of the composition g^(-1)(g(x)) is equal to |x|, which is the absolute value of x. This is not the same as the identity function, so it does not demonstrate that g^(-1)(x) and g(x) are inverses.

Therefore, only the composition g(g^(-1)(x)) = x shows that g(x) and g^(-1)(x) are inverses.