The function $f(x) = \frac{3x-4}{x+2}$ is a rational function. Let’s analyze it in detail:

1. Domain:

The domain is all real numbers except where the denominator is zero.

$x + 2 = 0 \implies x = -2$

So, the domain is: $x \in \mathbb{R}, \, x \neq -2$

2. Horizontal Asymptote:

Since the degree of the numerator and denominator are the same (both are degree 1), the horizontal asymptote is determined by the ratio of the leading coefficients: $\text{Horizontal Asymptote: } y = \frac{3}{1} = 3$

3. Vertical Asymptote:

Vertical asymptotes occur where the denominator equals zero, which is at: $x = -2$

4. X-Intercept:

To find the $x$-intercept(s), set $f(x) = 0$ (numerator = 0): $3x - 4 = 0 \implies x = \frac{4}{3}$

5. Y-Intercept:

To find the $y$-intercept, set $x = 0$: $f(0) = \frac{3(0) - 4}{0 + 2} = \frac{-4}{2} = -2$

Summary:

• Domain: $x \in \mathbb{R}, \, x \neq -2$
• Horizontal Asymptote: $y = 3$
• Vertical Asymptote: $x = -2$
• X-Intercept: $x = \frac{4}{3}$
• Y-Intercept: $y = -2$

Would you like me to graph the function or explore further details?

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Related Questions:

1. How do you find the range of $f(x)$?
2. What happens to $f(x)$ as $x \to -\infty$ or $x \to \infty$?
3. How would you find the derivative of $f(x)$ for rates of change?
4. Could you solve $f(x) = 5$ to find specific values of $x$?
5. What is the significance of the asymptotes in real-world applications?

Tip:

For rational functions, always check both vertical and horizontal asymptotes to understand their behavior at extremes!