https://file.aiquickdraw.com/mathbot/file/1729633359075afrnw5sz.jpg

Let's address the mathematical questions from the image you provided. The function given is:

\[
f(x) = \frac{6x}{x^2 + 9}
\]

Now, I'll go through the answers and solutions step-by-step:

### (a) **Is the point \((- \sqrt{3}, 1)\) on the graph of \(f\)?**
To verify if this point is on the graph, substitute \(x = - \sqrt{3}\) into the function and check if the result equals 1 (the y-coordinate):

\[
f(- \sqrt{3}) = \frac{6(- \sqrt{3})}{(- \sqrt{3})^2 + 9} = \frac{-6 \sqrt{3}}{3 + 9} = \frac{-6 \sqrt{3}}{12} = - \frac{\sqrt{3}}{2}
\]

This is not equal to 1, so the point \((- \sqrt{3}, 1)\) is **not on the graph** of the function.

### (b) **If \(x = 1\), what is \(f(x)\)?**
Substitute \(x = 1\) into the function:

\[
f(1) = \frac{6(1)}{1^2 + 9} = \frac{6}{1 + 9} = \frac{6}{10} = \frac{3}{5}
\]

Thus, \(f(1) = \frac{3}{5}\).

The point on the graph where \(x = 1\) is \(\left(1, \frac{3}{5}\right)\).

### (c) **If \(f(x) = 1\), what is \(x\)?**
We need to solve the equation \(f(x) = 1\):

\[
\frac{6x}{x^2 + 9} = 1
\]

Multiply both sides by \(x^2 + 9\):

\[
6x = x^2 + 9
\]

Rearrange the terms:

\[
x^2 - 6x + 9 = 0
\]

This simplifies to:

\[
(x - 3)^2 = 0
\]

Thus, \(x = 3\). Therefore, the point on the graph where \(f(x) = 1\) is \((3, 1)\).

### (d) **What is the domain of \(f\)?**
The domain of a rational function is all real numbers except where the denominator is zero. The denominator is:

\[
x^2 + 9
\]

Since \(x^2 + 9\) is always positive for all real numbers, the domain of \(f(x)\) is:

\[
(-\infty, \infty)
\]

### (e) **List the x-intercepts, if any, of the graph of \(f\).**
The x-intercepts occur when \(f(x) = 0\). Set the numerator equal to 0:

\[
6x = 0
\]

Thus, \(x = 0\). Therefore, 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 \(x = 0\). Substitute \(x = 0\) into the function:

\[
f(0) = \frac{6(0)}{0^2 + 9} = 0
\]

Thus, the y-intercept is \((0, 0)\), which is the same as the x-intercept.

---

Let me know if you'd like more details or have any questions!

### Follow-up Questions:
1. How can we further analyze the critical points of the function \(f(x)\)?
2. What is the behavior of \(f(x)\) as \(x\) approaches infinity or negative infinity?
3. Can you find the derivative of \(f(x)\) to determine its increasing and decreasing intervals?
4. How would the graph of \(f(x)\) look like based on these intercepts and the domain?
5. What are the asymptotes of the function \(f(x)\)?

### Tip:
When dealing with rational functions, always remember to check for both domain restrictions and any vertical or horizontal asymptotes. These features help in sketching the graph and understanding the behavior of the function.

Answer the following questions about the function:
(a) Is the point (-√3, 1) on the graph of f?
(b) If x = 1, what is f(x)? What point is on the graph of f?
(c) If f(x) = 1, what is x? What point(s) is (are) on the graph of f?
(d) What is the domain of f?
(e) List the x-intercepts, if any, of the graph of f.
(f) List the y-intercept, if there is one, of the graph of f.

Explore the analysis of the function f(x) = 6x / (x^2 + 9). This problem addresses domain, range, intercepts, and function evaluation with radicals. Learn how to determine if points like (-√3, 1) are on the graph, solve equations involving radicals, and list intercepts.

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