find the domain, range, and all zeros/intercepts, if any, of the functions, 3/x-4

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?

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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.

Learn how to find the domain, range, and zeros of the function f(x) = 3/(x-4). Understand where the function is defined, its possible output values, and the presence of intercepts. Suitable for advanced high school students.

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