https://file.aiquickdraw.com/mathbot/file/1729333220478rua1f7oq.jpg

### (b) Domain and Range of the Function
- **Domain**: The function \( f(x) \) is defined for all values of \( x \) except where the denominator is zero. Thus, the domain is \( x \in \mathbb{R} \setminus \{3\} \) (all real numbers except \( x = 3 \)).
- **Range**: Since \( f(x) \) is always positive (being the reciprocal of a squared term), the range is \( f(x) > 0 \). Therefore, the range is \( (0, \infty) \).

### (c) Re-write the Function with Argument \( z \)
If we replace \( x \) with \( z \), the function becomes:
\[
f(z) = \frac{1}{(z - 3)^2}
\]

### (d) Use the Graph to Estimate \( f(1) \)
From the graph, we can observe that at \( x = 1 \), the value of \( f(x) \) appears to be around \( 0.25 \).

### (e) Calculate \( f(1) \) Exactly
To find \( f(1) \) exactly:
\[
f(1) = \frac{1}{(1 - 3)^2} = \frac{1}{(-2)^2} = \frac{1}{4}
\]

### (f) Modify the Function to Exclude \( x = 4 \) from the Domain
To omit \( x = 4 \) from the domain, we can define the function as:
\[
f(x) = \begin{cases}
\frac{1}{(x - 3)^2}, & x \neq 3 \text{ and } x \neq 4 \\
\text{undefined}, & x = 3 \text{ or } x = 4
\end{cases}
\]

Would you like further details on any part? Here are five questions related to this topic for further exploration:

1. What would happen to the graph if the function was \( f(x) = \frac{1}{(x - 3)^3} \)?
2. How does shifting the function \( f(x) = \frac{1}{(x - 3)^2} \) vertically affect the graph and range?
3. What is the significance of the vertical asymptote in this function's behavior?
4. How would you determine the horizontal asymptote of a rational function in general?
5. Can a rational function have more than one vertical asymptote? Why or why not?

**Tip**: When dealing with rational functions, always identify the points where the denominator becomes zero to find the vertical asymptotes and undefined points.

Consider the function f(x) = 1 / (x - 3)^2. (a) Plot the graph of the function. (b) Write down the domain and range of the function. (c) Re-write the function with argument z. (d) Use your graph to estimate f(1). (e) Use the function to calculate f(1) exactly. (f) Write down another function where x = 4 has to be omitted from the domain.

Explore the rational function f(x) = 1 / (x - 3)^2, its graph, domain, and range. Learn how to rewrite it with a new argument, estimate f(1), and calculate f(1) exactly. Suitable for advanced high school math students in grades 10-12.

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