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. Find the Inverse Function

**Given Function:**
$$ f(x) = \sqrt{x} $$

**To Find:**
The inverse function $$ g(x) $$ such that $$ f(g(x)) = x $$ and $$ g(f(x)) = x $$.

**Steps to Find the Inverse:**

1. **Replace $$ f(x) $$ with $$ y $$:**
   $$ y = \sqrt{x} $$

2. **Swap $$ x $$ and $$ y $$ to find the inverse:**
   $$ x = \sqrt{y} $$

3. **Solve for $$ y $$:**
   $$ x^2 = y $$
   $$ y = x^2 $$

**Inverse Function:**
$$ g(x) = x^2 $$

**Domain Consideration:**
- **Original Function $$ f(x) $$:** Defined for $$ x \geq 0 $$ because the square root of a negative number is not a real number.
- **Inverse Function $$ g(x) $$:** To maintain the inverse relationship, $$ g(x) $$ should also consider $$ x \geq 0 $$.

### 2. Verify That $$ f $$ and $$ g $$ Are Inverses

**Verification:**

1. **Check $$ f(g(x)) = x $$:**
   $$
   f(g(x)) = f(x^2) = \sqrt{x^2} = |x|
   $$
   Since we are considering $$ x \geq 0 $$, $$ |x| = x $$. Therefore,
   $$
   f(g(x)) = x
   $$

2. **Check $$ g(f(x)) = x $$:**
   $$
   g(f(x)) = g(\sqrt{x}) = (\sqrt{x})^2 = x
   $$

Both compositions $$ f(g(x)) $$ and $$ g(f(x)) $$ return $$ x $$ for $$ x \geq 0 $$, confirming that $$ f $$ and $$ g $$ are indeed inverse functions.

### **Conclusion**

- **Original Function:** $$ f(x) = \sqrt{x} $$
- **Inverse Function:** $$ g(x) = x^2 $$
- **Verification:** $$ f(g(x)) = x $$ and $$ g(f(x)) = x $$ for all $$ x \geq 0 $$.

This confirms that $$ f $$ and $$ g $$ are inverse functions of each other.

**Answer Summary:**
The inverse of f (x ) = √x is g (x ) = x². Verifying, f(g(x)) = √(x²) = x for all x ≥ 0. Thus, f and g are inverses.
The inverse of $$ f(x) = \sqrt{x} $$ is $$ g(x) = x^2 $$. Verifying, $$ f(g(x)) = x $$ for all $$ x \geq 0 $$. Therefore, $$ f $$ and $$ g $$ are inverse functions.

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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