Let's address each part of the question one by one for the function $f(x) = 3x^2 - x - 2$.

(a) Is the point $(-1, 2)$ on the graph of $f$?

To determine if the point $(-1, 2)$ is on the graph, substitute $x = -1$ into the function and see if $f(-1) = 2$:

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

Since $f(-1) = 2$, the point $(-1, 2)$ is indeed on the graph of $f(x)$.

(b) If $x = 2$, what is $f(x)$? What point is on the graph of $f$?

Substitute $x = 2$ into the function:

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

So, $f(2) = 8$. The point $(2, 8)$ is on the graph of $f(x)$.

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

Set $f(x) = -2$ and solve for $x$:

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

Adding 2 to both sides:

$3x^2 - x = 0$

Factor the equation:

$x(3x - 1) = 0$

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

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

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

The domain of a quadratic function is all real numbers since there are no restrictions on the values that $x$ can take. Thus, the domain of $f(x)$ is $(-\infty, \infty)$.

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

To find the $x$-intercepts, set $f(x) = 0$ and solve for $x$:

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

Use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where $a = 3$, $b = -1$, and $c = -2$:

$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-2)}}{2(3)}$

$x = \frac{1 \pm \sqrt{1 + 24}}{6} = \frac{1 \pm \sqrt{25}}{6} = \frac{1 \pm 5}{6}$

So:

$x = \frac{6}{6} = 1 \quad \text{or} \quad x = \frac{-4}{6} = -\frac{2}{3}$

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 $f$.

To find the $y$-intercept, set $x = 0$ and solve for $f(0)$:

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

The $y$-intercept is $(0, -2)$.

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Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do you determine the vertex of the quadratic function $f(x) = 3x^2 - x - 2$?
2. What is the axis of symmetry for the function $f(x) = 3x^2 - x - 2$?
3. How do you graph the function $f(x) = 3x^2 - x - 2$?
4. What are the maximum and minimum values of the function $f(x) = 3x^2 - x - 2$?
5. How does changing the coefficients of $x^2$, $x$, and the constant term affect the shape and position of the parabola?

Tip: When solving quadratic equations, always check if they can be factored easily before using the quadratic formula—it can save time!