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When working with rational functions like \(s(x) = \frac{x+2}{(x+3)(x-1)}\), finding intercepts can help you understand where the graph of the function will cross the axes.
To find the **x-intercepts**, set the numerator equal to zero and solve for \(x\). For our function, this means solving \(x + 2 = 0\), resulting in \(x = -2\). Hence, the x-intercept is at the point \((-2, 0)\). This is where the graph touches the x-axis.

The **y-intercept** is found by setting \(x = 0\) in the function and solving for \(s(x)\). Substituting zero gives \(s(0) = \frac{2}{-3}\), leading to the y-intercept at \((0, -\frac{2}{3})\). This point indicates where the graph crosses the y-axis. Spotting these intercepts is fundamental, as they provide essential markers on which the rest of your graph will build.

Asymptotes are lines that the graph of a rational function approaches but never quite touches.

Our rational function has two types of asymptotes: **vertical** and **horizontal**. **Vertical asymptotes** occur where the denominator of the function equals zero, provided the numerator isn't zero. For the function \(s(x)\), the denominator is \((x+3)(x-1)\). By setting this equal to zero, we find \(x = -3\) and \(x = 1\) as the points where vertical asymptotes occur. These lines vertically cross the x-axis and the graph will get infinitely close to them, but not touch.

Now considering the **horizontal asymptote**, for rational functions where the degree of the numerator is less than that of the denominator, the horizontal asymptote is \(y = 0\). In our function, the numerator has degree 1 while the denominator has degree 2. Therefore, the horizontal asymptote is at \(y = 0\). This line signifies the value the function will web towards but never hit as it extends towards infinity on either side.

The **domain** and **range** of a rational function describe the set of possible inputs (x-values) and outputs (y-values), respectively.

For a rational function such as \(s(x) = \frac{x+2}{(x+3)(x-1)}\), the domain is determined by where the function is undefined—typically where the denominator is zero. Here, \(x = -3\) and \(x = 1\) are such points. As these values make the function undefined, the domain consists of all real numbers except \(-3\) and \(1\). It is expressed in interval notation as \((-abla, -3) \cup (-3, 1) \cup (1, abla)\).

The **range** of the function involves understanding what y-values the function can take. With the horizontal asymptote at \(y = 0\), and knowing vertical asymptotes generally slice the y-values being achieved by the function, the graph of \(s(x)\) can approach but not include \(y = 0\). Similarly, the function can attain nearly all y-values except the horizontal asymptote, so the range is all real numbers except \(0\), written as \((-abla, 0) \cup (0, abla)\). Understanding these helps in visualizing how the function's values behave over intervals.