91Ó°ÊÓ

Understanding the domain and range of a function is essential for graphing it accurately. For the function \(f(x)=\frac{-3}{x-3}\), the domain is all real numbers except where the function is undefined. This happens when the denominator is zero. Setting the denominator equal to zero, we find \(x-3=0\), so \(x=3\) is excluded from the domain. Therefore, the domain of the function is all real numbers except \(x=3\).

To find the range, we look at the possible values of \(f(x)\). This rational function can take on any real value except it can never be zero because the numerator is a constant and never equals zero. Hence, the range is all real numbers except \(y=0\).

The domain and range can be summarized as follows:

• Domain: \( (-\text{∞}, 3) \bigcup (3, \text{∞}) \)
• Range: \( (-\text{∞}, 0) \bigcup (0, \text{∞}) \)

Asymptotes are lines that the graph of a function approaches but never actually touches. Identifying them is crucial for understanding the behavior of a rational function. For \(f(x)=\frac{-3}{x-3}\), there are two types of asymptotes: vertical and horizontal.

1. **Vertical Asymptote**: This occurs where the function is undefined, which is at \(x=3\). So, the vertical asymptote is the line \(x=3\). The graph will approach, but never touch or cross, this line.
2. **Horizontal Asymptote**: To find the horizontal asymptote, consider the behavior of \(f(x)\) as \(x\) approaches infinity or negative infinity. As \(x\) goes to ±∞, the value of the function approaches zero. So, the horizontal asymptote is the line \(y=0\).

In summary:

• Vertical Asymptote: \(x=3\)
• Horizontal Asymptote: \(y=0\)

Analyzing the behavior of a function near its asymptotes helps us understand how the function behaves as it approaches these lines.

For the vertical asymptote at \(x=3\):

• As \(x\) approaches 3 from the left (\(x \to 3^-\)), the function value \(f(x)\) becomes very large negatively, approaching negative infinity. This means the graph will descend steeply downwards as it gets close to \(x=3\) from the left side.
• As \(x\) approaches 3 from the right (\(x \to 3^+\)), the function value \(f(x)\) becomes very large positively, approaching positive infinity. Hence, the graph will ascend steeply upwards as it gets close to \(x=3\) from the right side.

Near the horizontal asymptote at \(y=0\):

• As \(x\) moves far to the right (\(x \to +\text{∞}\)) or far to the left (\(x \to -\text{∞}\)), the value of the function \(f(x)\) gets closer and closer to zero.

The graph gets closer to the \(y=0\) line but never actually touches it.

Plotting key points on the graph helps to get a clearer picture of the shape and positioning of the function. Here are some useful points to plot for \(f(x)=\frac{-3}{x-3}\):

Calculate a few values of \(f(x)\) at different \(x\)-values:

• For \(x=1\): \(f(1)=\frac{-3}{1-3}=\frac{-3}{-2}=1.5\)
• For \(x=5\): \(f(5)=\frac{-3}{5-3}=\frac{-3}{2}=-1.5\)
• For \(x=0\): \(f(0)=\frac{-3}{0-3}=1\)
• For \(x=-1\): \(f(-1)=\frac{-3}{-1-3}=0.75\)

By plotting these points and considering the asymptotes, you can now sketch the graph. Remember to show that the graph approaches the vertical asymptote (\(x=3\)) sharply, and tapers off approaching the horizontal asymptote (\(y=0\)). This ensures an accurate representation of the function's behavior.