Given the function $$ f(x) = x^{1/2} $$, find its inverse and then verify that the two functions are inverses by showing $$ f(g(x)) = x $$.

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To find the inverse of the function $$ f(x) = \sqrt{x} $$ and verify that they are indeed inverses, follow these steps:

### 1. **Finding the Inverse Function**

To find the inverse function $$ g(x) $$ of $$ f(x) $$, we need to solve for $$ x $$ in terms of $$ y $$ where $$ y = f(x) $$.

1. **Start with the original function:**
   $$
   y = \sqrt{x}
   $$

2. **Solve for $$ x $$:**
   $$
   y = \sqrt{x} \implies y^2 = x
   $$

3. **Express the inverse function:**
   $$
   g(y) = y^2
   $$

So, the inverse function is:
   $$
   g(x) = x^2
   $$

**Note:** Since the original function $$ f(x) = \sqrt{x} $$ is defined for $$ x \geq 0 $$, the inverse function $$ g(x) = x^2 $$ should also be restricted to $$ x \geq 0 $$ to ensure it is a true inverse (i.e., it passes the horizontal line test).

### 2. **Verifying That $$ f $$ and $$ g $$ Are Inverses**

To verify that $$ f $$ and $$ g $$ are inverses, we need to show that:
$$
f(g(x)) = x \quad \text{and} \quad g(f(x)) = x
$$

Let's verify $$ f(g(x)) = x $$:

1. **Compose $$ f $$ and $$ g $$:**
   $$
   f(g(x)) = f(x^2)
   $$

2. **Apply the definition of $$ f $$:**
   $$
   f(x^2) = \sqrt{x^2}
   $$

3. **Simplify:**
   $$
   \sqrt{x^2} = |x|
   $$
   
   However, since $$ g(x) = x^2 $$ was defined for $$ x \geq 0 $$, $$ |x| = x $$ in this domain. Therefore:
   $$
   f(g(x)) = x \quad \text{for} \quad x \geq 0
   $$

Similarly, let's verify $$ g(f(x)) = x $$:

1. **Compose $$ g $$ and $$ f $$:**
   $$
   g(f(x)) = g(\sqrt{x})
   $$

2. **Apply the definition of $$ g $$:**
   $$
   g(\sqrt{x}) = (\sqrt{x})^2
   $$

3. **Simplify:**
   $$
   (\sqrt{x})^2 = x
   $$

Therefore:
$$
g(f(x)) = x \quad \text{for} \quad x \geq 0
$$

### 3. **Conclusion**

The inverse function of $$ f(x) = \sqrt{x} $$ is $$ g(x) = x^2 $$ defined for $$ x \geq 0 $$. We have verified that:
$$
f(g(x)) = \sqrt{x^2} = x \quad \text{and} \quad g(f(x)) = (\sqrt{x})^2 = x
$$
for all $$ x \geq 0 $$. This confirms that $$ f $$ and $$ g $$ are indeed inverse functions of each other.

**Final Answer:**
The inverse of $$ f(x) = \sqrt{x} $$ is $$ g(x) = x^2 $$ for $$ x \geq 0 $$. Verifying their inverse relationship:
$$
f(g(x)) = \sqrt{x^{2}} = x \quad \text{for all} \quad x \geq 0
$$
The inverse of $$ f(x) = \sqrt{x} $$ is $$ g(x) = x^2 $$ for $$ x \geq 0 $$.

Given the function \( f(x) = x^{1/2} \), find its inverse and then verify that the two functions are inverses by showing \( f(g(x)) = x \).

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