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Understanding the domain of a function is an important starting point for analyzing any function, especially rational functions. The domain consists of all the possible values that can be input into the function without causing any mathematical errors or undefined behaviors.
For rational functions, these are usually any values that don't make the denominator zero. If the denominator is zero, the function is undefined at that point.
In our exercise with the function \(P(x) = \frac{1 - 3x}{1 - x}\):

• Set the denominator \(1-x=0\) to find where the function is undefined.
• Solve for \(x\) to get \(x=1\). This means \(x=1\) is excluded from the domain.

Therefore, the domain of the function \(P(x)\) is all real numbers except \(x=1\). Understanding the domain helps us know where to plot the function and where to expect potential discontinuities.

Intercepts are points where the graph of the function crosses the axes. These are important reference points for sketching and understanding the function's behavior.

• **X-Intercept**: This is where the graph crosses the x-axis. To find it, set the numerator equal to zero because at the intercept, \(P(x) = 0\).
• **Y-Intercept**: This is where the graph crosses the y-axis. To find it, substitute \(x = 0\) into the function \(P(x)\).

For the function \(P(x) = \frac{1 - 3x}{1 - x}\):- To find the x-intercept, solve \(1 - 3x = 0\), giving \(x = \frac{1}{3}\). This means the graph crosses the x-axis at \(x = \frac{1}{3}\).- For the y-intercept, substitute \(x = 0\) into the function, resulting in \(P(0) = \frac{1}{1} = 1\). This gives us the point \((0, 1)\).
Finding these intercepts makes it easier to sketch the graph, as they serve as key markers.

Asymptotes are lines that the graph of a function approaches but never touches or crosses. They paint a clear picture of the function's behavior as it extends towards infinity or in the case of vertical asymptotes, at specific points.

• **Vertical Asymptote**: Occurs where the function is undefined. For \(P(x) = \frac{1 - 3x}{1 - x}\), set the denominator \(1 - x = 0\).
• **Horizontal Asymptote**: Determined by comparing the degrees of the numerator and denominator.

- For the vertical asymptote, solving \(1 - x = 0\) finds \(x = 1\).- For the horizontal asymptote, since both numerator and denominator are of degree 1, we look at the coefficients: \(\frac{-3}{-1} = 3\), so the horizontal asymptote is \(y=3\).
Including asymptotes in the graph helps define its boundaries and aids in understanding how the function behaves at extreme values.

Once we know the domain, intercepts, and asymptotes of a function, we can begin to sketch the graph.
Rational functions, like \(P(x) = \frac{1-3x}{1-x}\), often have complex shapes, so these identifiers become our guide.

• **Plot Known Points**: Start by plotting any intercepts and marking the asymptotes.
• **Evaluate Additional Points**: Choose some points within the domain to further refine the curve.

Steps to plot:1. Mark the vertical asymptote at \(x=1\), and the horizontal asymptote at \(y=3\).2. Place the x-intercept at \((\frac{1}{3}, 0)\) and the y-intercept at \((0, 1)\).3. To fill in the graph, calculate \(P(x)\) for other values of \(x\) to understand the curve's path.
Graphing becomes easier as it transforms abstract equations into visible shapes, making it clear where the function increases or decreases and its overall behavior.