Given the function $f(x) = 3x^2 - x - 2$, let's answer each question step by step:

(a) Is the point $(2, 8)$ on the graph of $f$?

To check if the point $(2, 8)$ is on the graph, we need to evaluate $f(2)$ and see if the result equals 8.

$f(2) = 3(2)^2 - (2) - 2 = 3(4) - 2 - 2 = 12 - 2 - 2 = 8$

Since $f(2) = 8$, the point $(2, 8)$ is indeed on the graph of $f$.

(b) If $x = -1$, what is $f(x)$? What point is on the graph of $f$?

We need to calculate $f(-1)$:

$f(-1) = 3(-1)^2 - (-1) - 2 = 3(1) + 1 - 2 = 3 + 1 - 2 = 2$

So, $f(-1) = 2$, which means the point $(-1, 2)$ is on the graph of $f$.

(c) If $f(x) = -2$, what is $x$? What point(s) are on the graph of $f$?

We solve for $x$ when $f(x) = -2$:

$3x^2 - x - 2 = -2$

Add 2 to both sides:

$3x^2 - x = 0$

Factor the equation:

$x(3x - 1) = 0$

So, $x = 0$ or $3x - 1 = 0$, which gives $x = \frac{1}{3}$.

Thus, the points $(0, -2)$ and $\left( \frac{1}{3}, -2 \right)$ are on the graph of $f$.

(d) What is the domain of $f$?

The function $f(x) = 3x^2 - x - 2$ is a polynomial, and polynomials are defined for all real numbers. Therefore, the domain of $f$ is:

$\text{Domain of } f = (-\infty, \infty)$

(e) List the $x$-intercept(s), if any, of the graph of $f$.

The $x$-intercepts occur when $f(x) = 0$:

$3x^2 - x - 2 = 0$

We solve this quadratic equation using the quadratic formula:

$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-2)}}{2(3)} = \frac{1 \pm \sqrt{1 + 24}}{6} = \frac{1 \pm \sqrt{25}}{6} = \frac{1 \pm 5}{6}$

This gives two solutions:

$x = \frac{1 + 5}{6} = 1 \quad \text{and} \quad x = \frac{1 - 5}{6} = -\frac{2}{3}$

Therefore, the $x$-intercepts are $(1, 0)$ and $\left( -\frac{2}{3}, 0 \right)$.

(f) List the $y$-intercept, if there is one, of the graph of $f$.

The $y$-intercept occurs when $x = 0$:

$f(0) = 3(0)^2 - 0 - 2 = -2$

So, the $y$-intercept is $(0, -2)$.

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Do you want more details on any step? Here are five related questions you might want to explore:

1. What is the vertex of the function $f(x) = 3x^2 - x - 2$?
2. How would you find the axis of symmetry for this quadratic function?
3. What is the range of the function $f(x) = 3x^2 - x - 2$?
4. Can you sketch the graph of the function using the points we found?
5. What is the behavior of $f(x)$ as $x \to \infty$ and $x \to -\infty$?

Tip: For quadratic functions, the vertex can be found using the formula $x = -\frac{b}{2a}$, where $a$ and $b$ come from the standard form $ax^2 + bx + c$.