https://file.aiquickdraw.com/mathbot/file/1730816056766gv589ivp.jpg

Let's focus on question number 2, where the function given is:

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

To sketch the graph of this function, let's analyze its properties step-by-step:

### 1. Domain
The function \( f(x) = \frac{1}{x^2 + 1} \) is defined for all real values of \( x \) because \( x^2 + 1 \) is always positive (it's never zero). Therefore, the domain is all real numbers, \( x \in \mathbb{R} \).

### 2. Range
Since \( x^2 + 1 \geq 1 \) for all real \( x \), \( f(x) \) will always be between 0 and 1:
\[
0 < f(x) \leq 1
\]
Thus, the range of the function is \( 0 < f(x) \leq 1 \).

### 3. Asymptotes
There is no vertical asymptote because the function is defined for all \( x \). However, as \( x \) approaches infinity or negative infinity, \( f(x) \) approaches 0. This means there is a horizontal asymptote at \( y = 0 \).

### 4. Symmetry
Since \( f(-x) = f(x) \), the function is even, meaning the graph is symmetric with respect to the y-axis.

### 5. Behavior and Key Points
- At \( x = 0 \): \( f(0) = \frac{1}{0^2 + 1} = 1 \).
- As \( x \to \infty \) or \( x \to -\infty \), \( f(x) \to 0 \).

### Sketch Summary
1. The graph has a maximum point at \( (0, 1) \).
2. It approaches the x-axis (y = 0) as \( x \) becomes very large in either direction.
3. The graph is symmetric about the y-axis.

Would you like further details on any of these steps or a graphical sketch of the function?

### Additional Questions
1. What is the derivative of \( f(x) = \frac{1}{x^2 + 1} \), and how does it affect the shape of the graph?
2. How does this function compare to \( f(x) = \frac{1}{x^2} \) in terms of asymptotes and behavior?
3. How would the range of \( f(x) = \frac{1}{x^2 + 1} \) change if the denominator were \( x^2 + 2 \)?
4. What would be the effect of adding a constant to the function, like \( f(x) = \frac{1}{x^2 + 1} + 1 \)?
5. How does the symmetry of \( f(x) = \frac{1}{x^2 + 1} \) help in sketching the graph quickly?

### Tip
For rational functions like \( f(x) = \frac{1}{x^2 + 1} \), always check the domain and asymptotes first. This will provide a clear framework for sketching the graph.

Explore the steps to sketch the graph of the rational function f(x) = 1 / (x^2 + 1). Learn about its domain, range, asymptotes, and symmetry. This guide is ideal for high school students studying rational functions and graph analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation