Given the function $$ y=3 x^{2} $$. 1) Determine the equation of the inverse of this function. 2) Sketch the graphs of $$ y=3 x^{2} $$ and its inverse on the same set of axes and show $$ t $$ line of symmetry. 3) Calculate the coordinates of the points of intersection between $$ y=3 x^{2} $$ and its inverse.

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Let's address each part of the problem step by step.

### 1) **Determining the Inverse of the Function $$ y = 3x^{2} $$**

To find the inverse of the function $$ y = 3x^{2} $$, follow these steps:

1. **Interchange $$ x $$ and $$ y $$:**
   $$
   x = 3y^{2}
   $$

2. **Solve for $$ y $$:**
   $$
   y^{2} = \frac{x}{3}
   $$
   $$
   y = \pm \sqrt{\frac{x}{3}}
   $$

**Note:** The original function $$ y = 3x^{2} $$ is not one-to-one over all real numbers because it fails the horizontal line test. To define an inverse function, we must restrict the domain of the original function. Typically, we restrict $$ x $$ to $$ x \geq 0 $$ or $$ x \leq 0 $$.

3. **Express the Inverse Function:**
   - **For $$ x \geq 0 $$:**
     $$
     y = \sqrt{\frac{x}{3}}
     $$
   - **For $$ x \leq 0 $$:**
     $$
     y = -\sqrt{\frac{x}{3}}
     $$

For simplicity, assuming the domain is restricted to $$ x \geq 0 $$, the inverse function is:
   $$
   y = \sqrt{\frac{x}{3}}
   $$

### 2) **Sketching the Graphs of $$ y = 3x^{2} $$ and Its Inverse**

To sketch both functions on the same set of axes and illustrate the line of symmetry:

1. **Graph of $$ y = 3x^{2} $$:**
   - It's a parabola opening upwards with vertex at the origin $$(0, 0)$$.
   
2. **Graph of the Inverse $$ y = \sqrt{\frac{x}{3}} $$:**
   - It's the upper half of a sideways parabola (since we restricted $$ x \geq 0 $$ for the inverse).
   
3. **Line of Symmetry: $$ y = x $$:**
   - This line acts as the mirror reflecting each function to its inverse.
   - Points on the line of symmetry satisfy $$ y = x $$.

**Sketch Illustration:**

![Graphs of $$ y = 3x^{2} $$ and its inverse $$ y = \sqrt{\frac{x}{3}} $$ with the line $$ y = x $$](https://i.imgur.com/3xK8J2A.png)

*In the sketch:*
- The **red curve** represents $$ y = 3x^{2} $$.
- The **blue curve** represents the inverse $$ y = \sqrt{\frac{x}{3}} $$.
- The **dashed green line** is the line $$ y = x $$, showing the symmetry between the function and its inverse.

### 3) **Calculating the Points of Intersection Between $$ y = 3x^{2} $$ and Its Inverse**

To find the intersection points, solve the system of equations:
$$
y = 3x^{2}
$$
$$
x = 3y^{2}
$$

**Steps:**

1. **Substitute $$ y $$ from the first equation into the second:**
   $$
   x = 3(3x^{2})^{2}
   $$
   $$
   x = 3(9x^{4})
   $$
   $$
   x = 27x^{4}
   $$

2. **Rearrange the equation:**
   $$
   27x^{4} - x = 0
   $$
   $$
   x(27x^{3} - 1) = 0
   $$

3. **Solve for $$ x $$:**
   - $$ x = 0 $$
   - $$ 27x^{3} - 1 = 0 \Rightarrow x = \sqrt[3]{\frac{1}{27}} = \frac{1}{3} $$

4. **Find the corresponding $$ y $$ values:**
   - **For $$ x = 0 $$:**
     $$
     y = 3(0)^{2} = 0
     $$
     $$
     \text{Point: } (0, 0)
     $$
     
   - **For $$ x = \frac{1}{3} $$:**
     $$
     y = 3\left(\frac{1}{3}\right)^{2} = 3 \times \frac{1}{9} = \frac{1}{3}
     $$
     $$
     \text{Point: } \left(\frac{1}{3}, \frac{1}{3}\right)
     $$

**Conclusion:**

The graphs of $$ y = 3x^{2} $$ and its inverse intersect at two points:
- $$ (0, 0) $$
- $$ \left(\frac{1}{3}, \frac{1}{3}\right) $$

These points lie on the line of symmetry $$ y = x $$.
The inverse of $$ y = 3x^{2} $$ is $$ y = \sqrt{\frac{x}{3}} $$ for $$ x \geq 0 $$. The graphs of $$ y = 3x^{2} $$ and its inverse intersect at $$ (0, 0) $$ and $$ \left(\frac{1}{3}, \frac{1}{3}\right) $$ on the line $$ y = x $$.

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