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The domain of a function is the complete set of possible input values (usually represented by the variable \( x \)) that a function can accept without resulting in errors like division by zero or taking the square root of a negative number. For rational functions, which are ratios of polynomials, the domain is all real numbers except for the values that make the denominator zero.

In the function \( y = \frac{x - 3}{x + 2} \), to find the domain, look at the denominator \( x + 2 \). Set the denominator equal to zero and solve for \( x \):

• \( x + 2 = 0 \)
• \( x = -2 \)

Therefore, \( x eq -2 \). Thus, the domain of this function is all real numbers except \( x = -2 \). This means any real number can be used as an input into the function except \( -2 \), because it would cause division by zero.

Graphing a rational function involves identifying and plotting key points such as intercepts, holes, and asymptotes. For the rational function \( y = \frac{x - 3}{x + 2} \), start by finding the intercepts and asymptotes.

X-intercept: Set \( y = 0 \) and solve for \( x \):

• \( \frac{x - 3}{x + 2} = 0 \) implies \( x - 3 = 0 \)
• Thus, \( x = 3 \) is the x-intercept.

Y-intercept: Set \( x = 0 \) and solve for \( y \):

• \( y = \frac{0 - 3}{0 + 2} = -\frac{3}{2} \)
• Thus, \( y = -\frac{3}{2} \) is the y-intercept.

Utilizing these intercepts along with the asymptote information, plot the curve and observe its approach near these critical points on the graph.

Asymptotes are lines that a graph approaches but never actually touches or crosses as it extends into infinity. These can be horizontal, vertical, or even oblique. For the function \( y = \frac{x - 3}{x + 2} \), focus on vertical and horizontal asymptotes.

Vertical Asymptote:

• Occurs where the denominator is zero and the numerator is not zero.
• For \( x + 2 = 0 \), \( x = -2 \) is a vertical asymptote.

Horizontal Asymptote:

• Compare degrees of the polynomials in the numerator and denominator.
• Since the degrees of both are 1 (linear), take the ratio of leading coefficients: \( \frac{1}{1} = 1 \).
• This gives a horizontal asymptote at \( y = 1 \).

These asymptotes are useful as guides when sketching the function graph, indicating where the function diverges or converges without bound.

Solving for \( y \) is about manipulating an equation so that \( y \) is on one side of the equation, usually to express \( y \) explicitly in terms of \( x \). Begin with an equation that may involve both variables. The goal is to isolate \( y \).

In our exercise \( xy + 2y = x - 3 \), to solve for \( y \):

• First, factor \( y \) from the left-hand side: \( y(x + 2) = x - 3 \).
• Next, divide both sides by \( x + 2 \): \( y = \frac{x - 3}{x + 2} \).

This makes \( y \) the subject of the formula, which is essential for graphing the function and understanding its dynamics such as domain and range, as well as intercepts and asymptotes.