To find the range of the function $f(x) = \frac{3x - 2}{x + 5}$, we need to determine all possible values that the function can take for different values of $x$.

Step 1: Identify Domain Restrictions

First, the function is undefined when the denominator is zero, i.e., when:

$x + 5 = 0 \implies x = -5$

Thus, the function is undefined at $x = -5$, and the domain is $x \in (-\infty, -5) \cup (-5, \infty)$.

Step 2: Analyze the Horizontal Asymptote

To find the horizontal asymptote, consider the behavior of the function as $x \to \infty$ or $x \to -\infty$. Divide both the numerator and denominator by $x$:

$f(x) = \frac{3x - 2}{x + 5} = \frac{3 - \frac{2}{x}}{1 + \frac{5}{x}}$

As $x \to \infty$ or $x \to -\infty$, the terms $\frac{2}{x}$ and $\frac{5}{x}$ approach zero, so:

$f(x) \to \frac{3}{1} = 3$

Thus, the function has a horizontal asymptote at $y = 3$.

Step 3: Check for Critical Points

To find the critical points, set $f(x) = y$ and solve for $x$ in terms of $y$:

$y = \frac{3x - 2}{x + 5}$

Multiply both sides by $x + 5$:

$y(x + 5) = 3x - 2$

Expand both sides:

$yx + 5y = 3x - 2$

Now, isolate $x$:

$yx - 3x = -5y - 2$

Factor out $x$:

$x(y - 3) = -5y - 2$

Solve for $x$:

$x = \frac{-5y - 2}{y - 3}$

Step 4: Exclude Values for Which $x = -5$

The function is undefined at $x = -5$. Set the denominator equal to zero to find which $y$-value is not possible:

$y - 3 = 0 \implies y = 3$

Thus, $y = 3$ is not in the range, since $f(x) \to 3$ as $x \to \infty$, but it never equals 3.

Conclusion: The Range

The range of the function is all real numbers except $y = 3$:

$\boxed{(-\infty, 3) \cup (3, \infty)}$

Would you like more details on this process?

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Here are 5 related questions to explore further:

1. What is the domain of the function $f(x) = \frac{3x - 2}{x + 5}$?
2. How do you find the vertical asymptote of a rational function?
3. How do you graph a rational function and identify its asymptotes?
4. What is the general method to find the range of any rational function?
5. How does the range change if the numerator and denominator have higher degrees?

Tip: Always check for asymptotes (vertical and horizontal) when analyzing rational functions, as they provide insight into the function's range and behavior.