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

pythonCopy code
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.