https://file.aiquickdraw.com/mathbot/file/17296330847238ngy9hrj.jpg

Let's analyze the image step by step and solve the questions regarding the function \( f(x) = \frac{6x^2}{x^2 + 9} \).

### Questions from the image:

#### (a) Is the point \((- \sqrt{3}, 1)\) on the graph of \(f\)?
We need to substitute \(x = -\sqrt{3}\) into the function and check if the result equals 1.

\[
f(-\sqrt{3}) = \frac{6(-\sqrt{3})^2}{(-\sqrt{3})^2 + 9} = \frac{6(3)}{3 + 9} = \frac{18}{12} = 1
\]
Yes, the point \((- \sqrt{3}, 1)\) is on the graph of \(f\).

#### (b) If \(x = 1\), what is \(f(x)\)? What point is on the graph of \(f\)?
Substitute \(x = 1\) into the function:
\[
f(1) = \frac{6(1)^2}{(1)^2 + 9} = \frac{6(1)}{1 + 9} = \frac{6}{10} = \frac{3}{5}
\]
So, the point on the graph when \(x = 1\) is \((1, \frac{3}{5})\).

#### (c) If \(f(x) = 1\), what is \(x\)? What point(s) are on the graph of \(f\)?
Set \(f(x) = 1\) and solve for \(x\):
\[
\frac{6x^2}{x^2 + 9} = 1
\]
Multiply both sides by \(x^2 + 9\):
\[
6x^2 = x^2 + 9
\]
Rearrange the equation:
\[
6x^2 - x^2 = 9 \implies 5x^2 = 9 \implies x^2 = \frac{9}{5}
\]
Thus:
\[
x = \pm \sqrt{\frac{9}{5}} = \pm \frac{3}{\sqrt{5}} = \pm \frac{3\sqrt{5}}{5}
\]
The points on the graph are \(\left(\frac{3\sqrt{5}}{5}, 1\right)\) and \(\left(-\frac{3\sqrt{5}}{5}, 1\right)\).

#### (d) What is the domain of \(f\)?
Since the function is a rational function with no restrictions (no values that make the denominator zero), the domain is all real numbers:
\[
\text{Domain of } f(x): (-\infty, \infty)
\]

#### (e) List the x-intercepts, if any, of the graph of \(f\).
To find the x-intercepts, set \(f(x) = 0\):
\[
\frac{6x^2}{x^2 + 9} = 0
\]
This implies that \(6x^2 = 0\), which results in \(x = 0\).
Thus, the x-intercept is \((0, 0)\).

#### (f) List the y-intercept, if there is one, of the graph of \(f\).
The y-intercept occurs when \(x = 0\):
\[
f(0) = \frac{6(0)^2}{(0)^2 + 9} = \frac{0}{9} = 0
\]
Thus, the y-intercept is \((0, 0)\).

---

Do you need more details on any part of this explanation?

### 5 Related Questions:
1. How do we find vertical and horizontal asymptotes of this rational function?
2. Can you explain how to calculate the derivative of \(f(x)\) to find critical points?
3. How does the behavior of the function change as \(x\) approaches infinity or negative infinity?
4. What is the range of the function \(f(x)\)?
5. How would you graph the function and its key points?

### Tip:
For rational functions, always check for asymptotes by analyzing the denominator and limits of the function as \(x\) approaches large values.

The function f(x) = (6x^2) / (x^2 + 9) is analyzed, with questions involving the point (-√3, 1), domain, intercepts, and calculations for specific values of x.

Explore the detailed analysis of the rational function f(x) = (6x^2) / (x^2 + 9). Solve for specific points, intercepts, and the domain of the function, ideal for students in Grades 9-12.

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