https://file.aiquickdraw.com/mathbot/file/17274173210510n0dlabh.jpg

The function given in the image is:

\[
f(x) = \frac{4x - 3}{x - 5}
\]

### 1. **Domain**:
The domain of a function refers to all possible values of \( x \) for which the function is defined. Since the function has a denominator, the value of \( x \) that makes the denominator zero must be excluded from the domain.

Here, the denominator is \( x - 5 \). Setting this equal to zero:

\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]

So, \( x = 5 \) is not allowed. Thus, the domain is:

\[
\text{Domain: } (-\infty, 5) \cup (5, \infty)
\]

### 2. **Range**:
To find the range, we analyze the behavior of the function. Since this is a rational function, we look at the asymptotic behavior and consider the values that \( f(x) \) can approach but not attain.

As \( x \to \infty \) or \( x \to -\infty \), the function behaves like:

\[
f(x) \approx \frac{4x}{x} = 4
\]

Thus, the function approaches but never reaches \( y = 4 \). Therefore, the range is all real numbers except \( y = 4 \):

\[
\text{Range: } (-\infty, 4) \cup (4, \infty)
\]

### 3. **Asymptotes**:
There are two types of asymptotes to consider: vertical and horizontal.

- **Vertical Asymptote**: This occurs where the denominator is zero and the function tends to infinity. Setting \( x - 5 = 0 \), we get \( x = 5 \). Hence, the vertical asymptote is:

\[
x = 5
\]

- **Horizontal Asymptote**: For large values of \( x \), the function behaves like:

\[
f(x) \approx \frac{4x}{x} = 4
\]

Thus, the horizontal asymptote is:

\[
y = 4
\]

### Summary:
- **Domain**: \( (-\infty, 5) \cup (5, \infty) \)
- **Range**: \( (-\infty, 4) \cup (4, \infty) \)
- **Vertical Asymptote**: \( x = 5 \)
- **Horizontal Asymptote**: \( y = 4 \)

Would you like further details or clarifications?

### Related Questions:
1. What happens to the function as \( x \) approaches the vertical asymptote?
2. How does the function behave around the horizontal asymptote for very large positive and negative values of \( x \)?
3. Can you graph the function based on the domain, range, and asymptotes?
4. How does the function change if the numerator or denominator is modified?
5. What other functions might have similar vertical or horizontal asymptotes?

**Tip**: To find horizontal asymptotes of rational functions, compare the degrees of the numerator and denominator.

Find the domain, range, and asymptotes for the function f(x) = (4x - 3) / (x - 5).

Explore how to find the domain, range, and asymptotes of the rational function f(x) = (4x - 3) / (x - 5). This detailed guide covers key concepts such as vertical and horizontal asymptotes, domain restrictions, and the behavior of rational functions. Suitable for students in Grades 9-12.

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