•Find the range for the following function: f(x)=1/(x-3)

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).

Find the range for the following function: f(x) = 1/(x - 3)

Learn how to find the range of the rational function f(x) = 1/(x - 3). This problem involves understanding asymptotic behavior, continuity, and identifying values where the function is undefined. Suitable for high school students in Grades 9-12.

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