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.