To graph the inverse of $y = 2^x$, let's follow the steps systematically.

Step 1: Write the Inverse Function

The original function is $y = 2^x$. To find the inverse of this function, we switch $x$ and $y$, and then solve for $y$:

$x = 2^y$

Now, take the logarithm base 2 of both sides to isolate $y$:

$\log_2(x) = y$

Thus, the inverse function is:

$y = \log_2(x)$

Step 2: Graph of $y = 2^x$ and its Inverse $y = \log_2(x)$

• The graph of $y = 2^x$ is an exponential curve that passes through $(0, 1)$ and rises rapidly as $x$ increases.
• The graph of $y = \log_2(x)$ is a logarithmic curve that passes through $(1, 0)$ and increases slowly as $x$ increases.

The inverse function reflects the original graph across the line $y = x$.

Step 3: Plotting the Graph

Let me graph both functions $y = 2^x$ and its inverse $y = \log_2(x)$.

I'll generate the graph now. numpy as np import matplotlib.pyplot as plt

Define the functions

def f(x): return 2**x

def f_inv(x): return np.log2(x)

Generate values for x

x_vals = np.linspace(-2, 4, 400) y_vals_f = f(x_vals) y_vals_inv = f_inv(x_vals_vals := np.linspace(0.01, 4, 400))

Create the plot

plt.figure(figsize=(8, 6)) plt.plot(x_vals, y_vals_f, label=r'$y = 2^x$', color='blue') plt.plot(x_vals_vals, y_vals_inv, label=r'$y = \log_2(x)$', color='green')

Plot the line y = x for reference

plt.plot(x_vals, x_vals, label=r'$y = x$', color='red', linestyle='--')

Labels and title

plt.title('Graph of $y = 2^x$ and its inverse $y = \log_2(x)$') plt.xlabel('x') plt.ylabel('y') plt.legend()

Show grid and plot

plt.grid(True) plt.xlim([-2, 4]) plt.ylim([-1, 4]) plt.axhline(0, color='black',linewidth=1) plt.axvline(0, color='black',linewidth=1) plt.show()Here's the graph showing both the function $y = 2^x$ (blue) and its inverse $y = \log_2(x)$ (green). The red dashed line represents the line $y = x$, which serves as the line of reflection between the two curves.

• The blue curve rises steeply as $x$ increases.
• The green curve starts from $(1, 0)$ and increases gradually, representing the logarithmic function.

Would you like further clarification or additional details about how the graphs relate?

Here are 5 related questions you might find interesting:

1. How does the graph of $y = 3^x$ compare to $y = 2^x$?
2. What happens if the base of the exponential function is less than 1, such as in $y = (1/2)^x$?
3. How do logarithmic functions with different bases compare?
4. What is the domain and range of $y = 2^x$ and $y = \log_2(x)$?
5. How can logarithmic and exponential functions be used in real-life applications?

Tip: When graphing an inverse function, the original function and its inverse are reflections of each other across the line $y = x$.