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The image provided presents a problem involving the graphing of a rational function. Let's break down the requirements and solve the problem step by step.

### Problem Overview:
You're asked to graph the rational function \( f(x) = \frac{1}{x^2} \). The tasks include:
1. Finding the **inverse**.
2. Determining the **x-intercept**.
3. Identifying the **vertical asymptote**.
4. Identifying the **horizontal asymptote**.
5. Filling out the **Table of Values**.
6. Finally, **graphing** the function.

### Solution Steps:

#### a. **Inverse of the Function**
The inverse of a function \( f(x) \) is found by swapping \( x \) and \( y \) and solving for \( y \).

Given \( y = \frac{1}{x^2} \):
1. Swap \( x \) and \( y \): \( x = \frac{1}{y^2} \).
2. Solve for \( y \): \( y^2 = \frac{1}{x} \) so \( y = \pm\frac{1}{\sqrt{x}} \).

Thus, the inverse functions are \( y = \frac{1}{\sqrt{x}} \) and \( y = -\frac{1}{\sqrt{x}} \), but only for \( x > 0 \).

#### b. **X-Intercept**
To find the x-intercept, set \( y = 0 \) and solve for \( x \):

\[ 0 = \frac{1}{x^2} \]

However, \( \frac{1}{x^2} \) can never equal 0, so **there is no x-intercept**.

#### c. **Vertical Asymptote**
A vertical asymptote occurs where the function is undefined, which happens when the denominator is zero. For \( f(x) = \frac{1}{x^2} \), this occurs at \( x = 0 \). So, the vertical asymptote is:

\[ x = 0 \]

#### d. **Horizontal Asymptote**
As \( x \) approaches infinity or negative infinity, \( \frac{1}{x^2} \) approaches 0. Therefore, the horizontal asymptote is:

\[ y = 0 \]

#### e. **Table of Values**
Let's calculate the values for the given x-values:
\[
\begin{array}{c|c}
x & f(x) = \frac{1}{x^2} \\
\hline
-3 & \frac{1}{(-3)^2} = \frac{1}{9} \approx 0.111 \\
-2 & \frac{1}{(-2)^2} = \frac{1}{4} = 0.25 \\
-1 & \frac{1}{(-1)^2} = \frac{1}{1} = 1 \\
0 & \text{Undefined} \\
1 & \frac{1}{(1)^2} = 1 \\
2 & \frac{1}{(2)^2} = \frac{1}{4} = 0.25 \\
3 & \frac{1}{(3)^2} = \frac{1}{9} \approx 0.111 \\
\end{array}
\]

#### f. **Graphing**
To graph the function \( f(x) = \frac{1}{x^2} \):

1. Plot the calculated points.
2. Notice the vertical asymptote at \( x = 0 \).
3. The curve is symmetric about the y-axis (even function).
4. As \( x \) moves away from 0 in either direction, the function approaches the horizontal asymptote at \( y = 0 \).

You can now graph these points along with the asymptotes to complete the task.

### Additional Questions and Tips

1. **What happens to the graph as \( x \) approaches zero from either side?**
2. **How does the graph behave as \( x \) becomes very large?**
3. **If \( f(x) \) had been \( \frac{1}{x} \) instead, how would the graph differ?**
4. **How do the horizontal and vertical asymptotes help in sketching rational functions?**
5. **What other key characteristics are useful for graphing rational functions?**

**Tip:** When graphing rational functions, always start by identifying asymptotes and intercepts, as these guide the overall shape of the graph.

Would you like any additional details on any of these steps?

Graph the rational function f(x) = 1/x^2 and find the inverse, intercept, vertical asymptote, horizontal asymptote, and table of values.

Step-by-step guide to graphing the rational function f(x) = 1/x^2. Includes finding the inverse function, identifying vertical and horizontal asymptotes, determining the x-intercept, and completing the table of values. Ideal for Grades 10-12 students learning about rational functions and their properties.

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