Let's analyze the function $f(x) = \frac{3}{x - 4}$.

1. Domain:

The domain of a function consists of all the values of $x$ for which the function is defined. The function $f(x) = \frac{3}{x - 4}$ is undefined where the denominator equals zero.

Set the denominator equal to zero and solve for $x$:

$x - 4 = 0$

$x = 4$

So, the function is undefined at $x = 4$. Thus, the domain of $f(x)$ is:

$\text{Domain: } (-\infty, 4) \cup (4, \infty)$

2. Range:

The range of a function consists of all possible values of $f(x)$. For the function $f(x) = \frac{3}{x - 4}$, since it is a rational function where the numerator is a constant (3) and the denominator can take any real value (except zero), the function can approach any real number except zero.

Thus, the range is:

$\text{Range: } (-\infty, 0) \cup (0, \infty)$

3. Zeros/Intercepts:

To find the zeros of the function, we set $f(x) = 0$ and solve for $x$:

$\frac{3}{x - 4} = 0$

However, a fraction equals zero only when the numerator is zero. Here, the numerator is 3, which is never zero. Therefore, there are no zeros (or x-intercepts) for this function.

Y-Intercept:

To find the y-intercept, we evaluate $f(x)$ at $x = 0$:

$f(0) = \frac{3}{0 - 4} = \frac{3}{-4} = -\frac{3}{4}$

Thus, the y-intercept is:

$\text{Y-Intercept: } \left(0, -\frac{3}{4}\right)$

Summary:

• Domain: $(-\infty, 4) \cup (4, \infty)$
• Range: $(-\infty, 0) \cup (0, \infty)$
• Zeros/Intercepts:
  • No zeros (x-intercepts)
  • Y-intercept at $\left(0, -\frac{3}{4}\right)$

Would you like further details or have any questions?

---

Here are five related questions:

1. How do you determine the domain of a more complex rational function?
2. What happens to the graph of $f(x) = \frac{3}{x-4}$ near the point where the function is undefined?
3. How would the domain and range change if the function was $f(x) = \frac{3}{x-4} + 2$?
4. Can a rational function have more than one y-intercept? Why or why not?
5. What are asymptotes, and how do they relate to the function $f(x) = \frac{3}{x-4}$?

Tip: When finding the domain of a function, always look out for values that make the denominator zero or cause other undefined operations like square roots of negative numbers.