(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: