Let's analyze the function $f(x) = \frac{1}{(x - 7)^2}$.

Step 1: Domain of the function

The domain of a function consists of all possible values of $x$ for which the function is defined. The given function $f(x) = \frac{1}{(x - 7)^2}$ has a denominator $(x - 7)^2$, which cannot be zero because division by zero is undefined.

So, set the denominator equal to zero and solve for $x$:

$(x - 7)^2 = 0 \quad \Rightarrow \quad x = 7$

Thus, the function is undefined at $x = 7$. The domain of the function is all real numbers except $x = 7$:

$\text{Domain: } (-\infty, 7) \cup (7, \infty)$

Step 2: X-intercept

To find the x-intercept, we set $f(x) = 0$ and solve for $x$.

However, the expression $\frac{1}{(x - 7)^2}$ is always positive for any $x \neq 7$, so the function never equals zero. Therefore, there is no x-intercept.

Step 3: Y-intercept

To find the y-intercept, set $x = 0$ and evaluate $f(0)$:

$f(0) = \frac{1}{(0 - 7)^2} = \frac{1}{49}$

Thus, the y-intercept is $\left( 0, \frac{1}{49} \right)$.

Step 4: Graph of the function

The function $f(x) = \frac{1}{(x - 7)^2}$ is a rational function with a vertical asymptote at $x = 7$. As $x \to 7$, the function value grows to infinity. As $x$ moves away from 7, the function decays towards zero but never touches the x-axis.

The graph has the following characteristics:

• Vertical asymptote at $x = 7$.
• No x-intercepts.
• Y-intercept at $\left( 0, \frac{1}{49} \right)$.
• The graph is symmetric around $x = 7$ because $(x - 7)^2$ is symmetric.

Summary

• Domain: $(-\infty, 7) \cup (7, \infty)$
• X-intercept: None
• Y-intercept: $\left( 0, \frac{1}{49} \right)$

Would you like me to graph this function for you or provide further details?

Related Questions:

1. How do vertical asymptotes affect the graph of a rational function?
2. Can a rational function have both vertical and horizontal asymptotes?
3. What are the general properties of functions with squared terms in the denominator?
4. How do you find horizontal asymptotes of rational functions?
5. How do symmetry properties affect the shape of a graph?

Tip: Rational functions can have vertical asymptotes where their denominators equal zero. Always check the domain for such discontinuities!