find an explicit formula for $$ f^{-1} $$and use it to graph $$ f^{-1} $$,f and the line y=x on the same screen. to check your work , see whether the graphs of f and $$ f^{-1} $$ are reflections about the line . f(x)=$$ \sqrt{x^2+x}\text{       x}>0 $$

To find the explicit formula for the inverse function $ f^{-1}(x) $, we start with the given function:

\[ f(x) = \sqrt{x^2 + x} \quad \text{for} \; x > 0 \]

To find the inverse, follow these steps:

1. **Express the function in terms of $ y $:**

\[ y = \sqrt{x^2 + x} \]

2. **Solve for $ x $ in terms of $ y $:**

First, square both sides to eliminate the square root:

\[ y^2 = x^2 + x \]

This is a quadratic equation in terms of $ x $. Rearrange it to standard form:

\[ x^2 + x - y^2 = 0 \]

3. **Solve the quadratic equation for $ x $:**

Use the quadratic formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ with $ a = 1 $, $ b = 1 $, and $ c = -y^2 $:

\[ x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-y^2)}}{2(1)} \]
\[ x = \frac{-1 \pm \sqrt{1 + 4y^2}}{2} \]

Since $ x > 0 $, we take the positive solution:

\[ x = \frac{-1 + \sqrt{1 + 4y^2}}{2} \]

Thus, the inverse function $ f^{-1}(x) $ is:

\[ f^{-1}(x) = \frac{-1 + \sqrt{1 + 4x^2}}{2} \]

Now, let's graph $ f(x) $, $ f^{-1}(x) $, and the line $ y = x $ on the same screen to check whether the graphs of $ f $ and $ f^{-1} $ are reflections about the line $ y = x $.

We can use Python and matplotlib to plot these functions.

### Python Code for Plotting

```python
import numpy as np
import matplotlib.pyplot as plt

# Define the original function f(x) and its inverse f_inverse(x)
def f(x):
    return np.sqrt(x**2 + x)

def f_inverse(x):
    return (-1 + np.sqrt(1 + 4*x**2)) / 2

# Define the range of x values for plotting
x = np.linspace(0, 10, 400)

# Plot f(x), f_inverse(x), and the line y=x
plt.plot(x, f(x), label='$f(x) = \\sqrt{x^2 + x}$')
plt.plot(x, f_inverse(x), label='$f^{-1}(x) = \\frac{-1 + \\sqrt{1 + 4x^2}}{2}$')
plt.plot(x, x, label='$y = x$', linestyle='dashed')

# Add labels and title
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graphs of $f(x)$, $f^{-1}(x)$, and $y=x$')
plt.legend()
plt.grid(True)
plt.show()
```

### Explanation
- The function $ f(x) = \sqrt{x^2 + x} $ is plotted.
- The inverse function $ f^{-1}(x) = \frac{-1 + \sqrt{1 + 4x^2}}{2} $ is plotted.
- The line $ y = x $ is plotted as a dashed line for reference.

By inspecting the plot, we can verify whether $ f $ and $ f^{-1} $ are reflections about the line $ y = x $.

Would you like details on any part of the process, or do you have any questions? Here are five questions you might consider:

1. How do we know which root to take when solving the quadratic equation for $ x $?
2. Can you explain why we square both sides when finding the inverse?
3. What happens if we try to find the inverse for $ x \leq 0 $?
4. How do we interpret the graph in terms of reflections about $ y = x $?
5. What are the limitations of this inverse function?

**Tip:** When finding the inverse of a function involving a square root, always consider the domain restrictions to ensure the solution is valid for the given context.

Learn how to find the inverse function of f(x) = sqrt(x^2 + x) and graph it alongside the original function and the line y = x. Explore the reflection properties of inverse functions with a step-by-step solution. Suitable for high school students.

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