The function you've provided is:

$f(x) = \frac{x + 3}{2x - 3}$

Let's analyze and explore this function in terms of its domain, potential asymptotes, and behavior.

1. Domain of the Function

The domain of a function includes all the values of $x$ for which the function is defined. Since the function involves a fraction, we need to ensure that the denominator is not zero, as division by zero is undefined.

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

$2x - 3 = 0$

$2x = 3$

$x = \frac{3}{2}$

Thus, $x = \frac{3}{2}$ would make the denominator zero, so the function is undefined at $x = \frac{3}{2}$. Therefore, the domain of $f(x)$ is all real numbers except $x = \frac{3}{2}$.

Domain: $x \in \mathbb{R}, \quad x \neq \frac{3}{2}$

2. Vertical Asymptote

A vertical asymptote occurs where the function goes to infinity or negative infinity as $x$ approaches a particular value. This happens when the denominator is zero, and the numerator is non-zero.

As $x$ approaches $\frac{3}{2}$ from either side, the function value will go to $+\infty$ or $-\infty$, depending on the direction of the approach.

Thus, there is a vertical asymptote at:

$x = \frac{3}{2}$

3. Horizontal Asymptote

Horizontal asymptotes describe the behavior of the function as $x$ approaches positive or negative infinity. For rational functions, the horizontal asymptote depends on the degrees of the polynomials in the numerator and denominator.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
• If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.

In $f(x) = \frac{x + 3}{2x - 3}$, the degree of the numerator and denominator is the same (both are linear, degree 1).

The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 2. Thus, the horizontal asymptote is:

$y = \frac{1}{2}$

4. Behavior of the Function

To fully understand the behavior, we might also consider:

• Intercepts: $f(x)$ will have a y-intercept when $x = 0$ and an x-intercept when the numerator equals zero.

  • Y-intercept: $f(0) = \frac{0 + 3}{2(0) - 3} = \frac{3}{-3} = -1$.
  • X-intercept: Set $x + 3 = 0$, so $x = -3$.

• Sign Analysis: Analyze the function's sign by testing intervals between the intercepts and asymptotes to determine when $f(x)$ is positive or negative.

Summary

• Domain: $x \in \mathbb{R}, x \neq \frac{3}{2}$
• Vertical Asymptote: $x = \frac{3}{2}$
• Horizontal Asymptote: $y = \frac{1}{2}$
• X-intercept: $x = -3$
• Y-intercept: $y = -1$

Would you like a deeper explanation on any of these points? Here are some follow-up questions:

1. How do you determine the vertical asymptote for more complex rational functions?
2. Why does the function have a horizontal asymptote at $y = \frac{1}{2}$?
3. How do intercepts help in sketching the graph of a function?
4. What changes would occur in the function if the numerator were different?
5. How can you graphically represent the function $f(x)$?

Tip: When analyzing rational functions, always start with the domain and identify any asymptotes, as they guide the behavior of the function significantly.