To analyze the functions presented in the image, I'll break down each function, find its domain and range, and describe the general approach for sketching their graphs. Let's examine each one:

1. $h(x) = \frac{1}{x^2 - 3}$

• Domain: The denominator cannot be zero, so we set $x^2 - 3 \neq 0$, which gives $x \neq \pm \sqrt{3}$. Thus, the domain is all real numbers except $x = \pm \sqrt{3}$.
• Range: Since $x^2 \geq 0$, $x^2 - 3$ can take values from $-3$ to $+\infty$. However, since it’s in the denominator, the range will be all real values except zero.
• Graph: This function has vertical asymptotes at $x = \pm \sqrt{3}$. The graph approaches zero as $x$ approaches infinity or negative infinity.

2. $i(x) = \frac{x^2 - 2x}{x - 1}$

• Domain: The denominator $x - 1 \neq 0$, so $x \neq 1$. The domain is all real numbers except $x = 1$.
• Range: This function simplifies to $x - 1$ (after canceling $x - 1$ in the numerator and denominator) for $x \neq 1$. Hence, the range is all real numbers.
• Graph: The function $i(x)$ will resemble the line $y = x - 1$ but will have a hole at $x = 1$.

3. $j(x) = \frac{x^2 - 4}{x^2 + 6}$

• Domain: There are no values of $x$ that make the denominator zero, so the domain is all real numbers.
• Range: The function’s values will depend on the behavior as $x \to \infty$. Since it’s a rational function with the same degree in the numerator and denominator, the horizontal asymptote is the ratio of the leading coefficients, which is $y = 1$.
• Graph: This function approaches $y = 1$ as $x$ becomes very large or very small.

4. $k(x) = \frac{x - 6}{x - 1}$

• Domain: The denominator $x - 1 \neq 0$, so $x \neq 1$. Thus, the domain is all real numbers except $x = 1$.
• Range: This function can take all real values except the value at the vertical asymptote.
• Graph: The graph has a vertical asymptote at $x = 1$ and resembles a line $y = 1$ with some transformations.

Would you like detailed step-by-step calculations, or do you have specific questions on any part?

Here are five related questions that could further deepen understanding:

1. What are vertical asymptotes, and how do they affect the domain of a function?
2. How do we simplify rational expressions like $i(x) = \frac{x^2 - 2x}{x - 1}$?
3. Why does $j(x) = \frac{x^2 - 4}{x^2 + 6}$ have a horizontal asymptote at $y = 1$?
4. How can we determine if a rational function has holes in its graph?
5. How does the degree of the numerator and denominator in a rational function affect its range?

Tip: For rational functions, examining the zeros of the numerator and the values that make the denominator zero is crucial to determine the domain, asymptotes, and possible holes in the graph.