Problem 43 - Algebra 2

Question

Graph each rational function. \(f(x)=\frac{x^{2}-5 x+4}{x-4}\)

Step-by-Step Solution

Answer

The graph is a line \(y = x - 1\) with a hole at \((4, 3)\).

Step 1: Determine Domain

The domain of the rational function is all real numbers except where the denominator is zero. So, set the denominator equal to zero and solve: \(x - 4 = 0\). Thus, \(x = 4\) is where the function is undefined. The domain is all real numbers except \(x = 4\).

Step 2: Simplify the Function

We need to check if the rational function can be simplified. Factor the numerator: \(x^2 - 5x + 4 = (x - 1)(x - 4)\). The function becomes \(f(x) = \frac{(x - 1)(x - 4)}{x - 4}\). Cancel out the \(x-4\) term in the numerator and denominator to get \(f(x) = x - 1\), for \(x eq 4\).

Step 3: Identify Asymptotes

Inspect the simplified function to find vertical and horizontal asymptotes. Since \(x=4\) was canceled out, it is a removable discontinuity (hole). This function has a hole at \(x = 4\), and no vertical asymptotes remain. Because the simplified function is linear, it has no horizontal asymptote.

Step 4: Locate and Plot the Hole

The hole at \(x = 4\) occurs in the graph of \(f(x) = x - 1\). To find its \(y\)-coordinate, substitute \(x = 4\) into the simplified equation: \(y = 4 - 1 = 3\). Plot a hole at the point \((4, 3)\).

Step 5: Graph the Simplified Function

Plot the graph of \(y = x - 1\). This is a straight line with a slope of 1 and a y-intercept of \(-1\). Draw the line, making sure to leave out the hole at \((4, 3)\).

Step 6: Review and Finalize the Graph

Double-check the graph for errors: ensure there is a hole at \((4, 3)\) and the straight line \(y = x - 1\) is accurately drawn. There should be no vertical or horizontal asymptotes in the final graph.

Key Concepts

Domain of a Function

The domain of a rational function is simply the set of all possible input values (or "x-values") that will not make the denominator zero. Why is that important? Because if the denominator is zero, the function becomes undefined. This is a common characteristic of rational functions. To find the domain, follow these steps:

• Look at the denominator of the function.
• Set the denominator equal to zero and solve the equation to find the values of x that make it zero.
• The domain is all real numbers except these solutions.

For the function given, \( f(x) = \frac{x^2 - 5x + 4}{x - 4} \), the denominator is \( x - 4 \). Setting this equal to zero, we find \( x = 4 \). Thus, the domain is all real numbers except \( x = 4 \). That means anywhere you see \( x = 4 \), the function does not exist.

Asymptotes

Asymptotes in rational functions are lines that the graph of the function approaches but never touches. They help us understand the behavior of the graph at certain values of x. There are two main types of asymptotes for rational functions:

• Vertical asymptotes occur where the denominator is zero and cannot be canceled out. In this exercise, there is a term \( x - 4 \) in both the numerator and the denominator, which can be canceled, indicating no vertical asymptote, but rather, a removable discontinuity at \( x = 4 \).
• Horizontal asymptotes help describe the end behavior of the function as \( x \) goes to infinity. They exist when the degree (highest power) of the numerator is less than or equal to the degree of the denominator. For the simplified function, \( f(x) = x - 1 \), being linear, indicates no horizontal asymptote is present.

Discontinuity

Discontinuities in rational functions are points where the function is not continuous, meaning there is either a hole or a jump in the graph at these points. In the case of the given function:

• The original function \( f(x) = \frac{x^2 - 5x + 4}{x - 4} \) has a term \( x - 4 \) in both the numerator and the denominator. By simplifying the function, these terms cancel out, leaving \( f(x) = x - 1 \). This cancellation indicates a removable discontinuity or a hole in the graph at \( x = 4 \).
• To find the exact location of the hole, substitute \( x = 4 \) into the simplified equation \( f(x) = x - 1 \), giving us the y-coordinate \( y = 3 \). Thus, there is a hole at the point \( (4, 3) \).

Discontinuities are easy to miss but crucial for accurately graphing rational functions.

Graphing

Graphing rational functions involves several critical steps, especially when the function simplifies and contains discontinuities like holes:

• First, determine any discontinuities or potential asymptotes. Here, we found a removable discontinuity (a hole) at \( x = 4 \).
• Second, simplify the function. For \( f(x) = \frac{(x - 1)(x - 4)}{x - 4} \), it simplifies to \( f(x) = x - 1 \). Remember, this is valid only for \( x ot= 4 \).
• Next, graph the simplified function, \( y = x - 1 \). This is a straight line with a slope of 1 and a y-intercept of \( -1 \).
• Finally, make sure to plot the hole at the point \( (4, 3) \), where the original function is undefined.

By following these steps, you ensure the graph accurately represents both the behavior and limitations of the function.