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The domain of a function, specifically a rational function, refers to all the possible values of \(x\) for which the function is defined. For rational functions, which are fractions where both the numerator and denominator are polynomials, the domain excludes any values of \(x\) that make the denominator zero.
To find these values, set the denominator equal to zero and solve for \(x\). These solutions are the values you must exclude, as division by zero is undefined.

For example, if the denominator is \(x - 3\), set \(x - 3 = 0\). Solving this yields \(x = 3\), which means \(x = 3\) is not part of the domain. This examination helps understand where a rational function is valid and where it might have vertical asymptotes or holes.

Intercepts are critical points on a graph where the function crosses the axes. Understanding intercepts provides a clear picture of the function's behavior and path.

**X-Intercept:**
To find the x-intercepts, set the numerator of the function to zero and solve for \(x\). This is because the x-intercept(s) occur where the output value \(y\), which is the entire rational expression, equals zero.

• If the rational function is \(\frac{x - 2}{x + 1}\), set \(x - 2 = 0\). Solving this gives \(x = 2\), meaning the graph crosses the x-axis at \((2,0)\).

**Y-Intercept:**
The y-intercept is found by evaluating the function at \(x = 0\). Plugging in this value gives \(f(0) = \frac{-2}{1} = -2\), so the y-intercept is at \((0,-2)\).
Intercepts are foundational in sketching the graph and providing the structure needed to visualize changes in the function's direction.

Asymptotes are lines that a graph approaches but never quite touches, guiding the overall direction of the curve.

**Vertical Asymptotes:**
These occur at the values of \(x\) that make the denominator zero and are not canceled by the numerator. To identify these, solve \(x + 1 = 0\) for the function \(\frac{x - 2}{x + 1}\). This gives \(x = -1\), which means there's a vertical asymptote at \(x = -1\). The graph will spike toward infinity near this line but will never cross it.

**Horizontal Asymptotes:**
These depend on the relationship between the degrees of the polynomials in the numerator and the denominator.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
• If the degrees are equal, divide the coefficients of the highest degree terms. For example, with equal degrees, co-efficients being 1 would lead to an asymptote at \(y=1\).
• If the numerator's degree is greater, there isn’t a horizontal asymptote but a possible slant asymptote.

Understanding these asymptotes helps in sketching a rational function accurately as they define the behavior near the poles and extremes.