https://file.aiquickdraw.com/mathbot/file/17360106982497p14c5u7.jpg

The given function is:

\[
f(x) = \sqrt{x + 1}
\]

Let's analyze this function and plot its graph:

### 1. **Domain of the Function:**
The square root function $ \sqrt{x} $ is only defined for $ x \geq 0 $. Therefore, $ x + 1 \geq 0 $ for this function to be defined. This means:
\[
x \geq -1
\]

Thus, the domain of $ f(x) $ is:
\[
x \in [-1, \infty)
\]

### 2. **Key Points to Plot:**
To graph $ f(x) $, we calculate some key points in the domain:
- For $ x = -1 $: $ f(-1) = \sqrt{-1 + 1} = 0 $
- For $ x = 0 $: $ f(0) = \sqrt{0 + 1} = 1 $
- For $ x = 3 $: $ f(3) = \sqrt{3 + 1} = 2 $
- For $ x = 8 $: $ f(8) = \sqrt{8 + 1} = 3 $

### 3. **Behavior:**
- The function starts at $ (-1, 0) $ and increases as $ x $ increases.
- The graph will be smooth and curved upward because $ f(x) $ is a square root function.

I will now plot the graph of this function. numpy as np
import matplotlib.pyplot as plt

# Define the function f(x) = sqrt(x + 1)
def f(x):
    return np.sqrt(x + 1)

# Generate x values in the domain [-1, 10]
x_values = np.linspace(-1, 10, 500)
y_values = f(x_values)

# Plot the graph
plt.figure(figsize=(8, 6))
plt.plot(x_values, y_values, label=r"$f(x) = \sqrt{x + 1}$", color="blue")
plt.axhline(0, color='black', linewidth=0.8, linestyle='--')
plt.axvline(0, color='black', linewidth=0.8, linestyle='--')
plt.scatter([-1, 0, 3, 8], [0, 1, 2, 3], color="red", label="Key Points") # Highlight key points
plt.title("Graph of $f(x) = \sqrt{x + 1}$")
plt.xlabel("x")
plt.ylabel("f(x)")
plt.legend()
plt.grid(alpha=0.4)
plt.show()Here is the graph of the function $ f(x) = \sqrt{x + 1} $. Key points, such as $ (-1, 0), (0, 1), (3, 2), (8, 3) $, are highlighted in red. The graph starts at $ x = -1 $ and increases smoothly as $ x $ grows.

### Key Observations:
1. The function is only defined for $ x \geq -1 $.
2. The graph is smooth and concave down, as is typical for square root functions.
3. The graph approaches infinity as $ x \to \infty $, but at a slower rate.

Would you like further clarification or details? Below are related questions to expand on this topic:

---

### Related Questions:
1. How do transformations (like $ f(x) = \sqrt{x - c} + d $) affect the graph of square root functions?
2. What happens to the graph if the function is $ f(x) = -\sqrt{x + 1} $?
3. Can you find the inverse of $ f(x) = \sqrt{x + 1} $ and plot it?
4. What is the range of $ f(x) = \sqrt{x + 1} $?
5. How can square root functions be used to model real-world scenarios?

---

### Tip:
When graphing functions with square roots, always check the domain carefully and plot key points for accuracy!

Plot the graph of the following function, f(x) = sqrt(x + 1), and also use negative values.

Learn how to plot the graph of the function f(x) = sqrt(x + 1). Explore its domain, key points, and graph behavior, starting at x = -1 and using negative values. This problem covers concepts in graphing square root functions, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation