To find the range of the function $f(x) = \frac{1}{x-3}$, let's analyze it step by step.

Step 1: Identify the domain

The function $f(x)$ is undefined when the denominator is zero. So, let's find the value of $x$ that makes the denominator zero: $x - 3 = 0 \quad \Rightarrow \quad x = 3$ Thus, the function is undefined at $x = 3$. The domain of $f(x)$ is $x \in (-\infty, 3) \cup (3, \infty)$.

Step 2: Analyze the behavior of the function

Now, let's understand how the function behaves as $x$ approaches different values:

• As $x$ approaches 3 from the left (i.e., $x \to 3^-$), $f(x)$ becomes increasingly negative, tending towards negative infinity: $f(x) \to -\infty$.
• As $x$ approaches 3 from the right (i.e., $x \to 3^+$), $f(x)$ becomes increasingly positive, tending towards positive infinity: $f(x) \to +\infty$.

So, the function tends to $-\infty$ as $x$ approaches 3 from the left, and tends to $+\infty$ as $x$ approaches 3 from the right.

Step 3: Determine the range

Since the function can take arbitrarily large positive and negative values as $x$ approaches 3 from either side, and it is continuous everywhere else in its domain, the function can take all real values except for one specific value.

There is no horizontal asymptote preventing $f(x)$ from taking any value other than 0. The function never equals 0 because there is no value of $x$ such that $\frac{1}{x-3} = 0$.

Thus, the range of $f(x)$ is all real numbers except 0: $\text{Range} = (-\infty, 0) \cup (0, \infty)$

Would you like further details on this, or do you have any questions?

Follow-up Questions:

1. How would the range change if the function were $f(x) = \frac{1}{x + 3}$?
2. What would be the domain and range of the function $f(x) = \frac{1}{x^2 - 9}$?
3. How do vertical asymptotes affect the range of rational functions in general?
4. What is the range of $f(x) = \frac{2}{x - 3}$?
5. Can you find the range of a similar function $g(x) = \frac{1}{x^2 - 4}$?

Tip:

When analyzing the range of rational functions, consider both the asymptotic behavior and points where the function is undefined (such as vertical asymptotes).