Question

Use a graphing utility to graph the function. Use the graph to determine any \(x\) -value(s) at which the function is not continuous. Explain why the function is not continuous at the \(x\) -value(s). \(k(x)=\frac{x-4}{x^{2}-5 x+4}\)

Short answer

The function \(k(x)=\frac{x-4}{x^{2}-5 x+4}\) is discontinuous at \(x=1\) and \(x=4\). This is because at these points, the denominator of the function becomes zero making the function undefined. These points of discontinuity are visible on the function's graph as vertical asymptotes.

Step-by-step solution

Find where the denominator is zero

First, let's find the values of \(x\) where the denominator of the rational function \(k(x)=\frac{x-4}{x^{2}-5 x+4}\) equals zero. This is because a rational function is undefined and hence, not continuous where its denominator is zero. To do this, set the denominator \(x^{2}-5x+4 = 0\). Solve the quadratic equation to find the roots.

Solving the Quadratic equation

The quadratic equation \(x^{2}-5x+4 = 0\) can be factored to \((x - 1)(x - 4) = 0\). Solving this equation, we get the roots of the equation to be \(x=1\) and \(x=4\). Therefore, The function becomes undefined and is discontinuous at \(x=1\) and \(x=4\).

Graph the function

Now we will graph the function \(k(x)=\frac{x-4}{x^{2}-5 x+4}\) using a graphing utility. On the graph, draw vertical dashed lines at \(x=1\) and \(x=4\), the points of discontinuity observed. We will see vertical asymptotes at these lines indicating the discontinuity of the function at \(x=1\) and \(x=4\).

Explain why the function is discontinuous at these points

The function is discontinuous at \(x=1\) and \(x=4\) because at these values of \(x\), the denominator of the function \(k(x)=\frac{x-4}{x^{2}-5x+4}\) becomes zero, and hence, the function becomes undefined. In the graph, this can be visualized as the function approaching infinity around \(x=1\) and \(x=4\), resulting in the appearance of vertical asymptotes.

Key concepts

Rational Functions

A rational function is a fraction of polynomials. Specifically, it is expressed as one polynomial (the numerator) divided by another polynomial (the denominator). For example, the function \(k(x)=\frac{x-4}{x^{2}-5x+4}\) is a rational function. The key characteristic of rational functions is that they can exhibit discontinuities, which occur where the denominator equals zero, causing the function to be undefined.
Rational functions often have vertical asymptotes or holes at points of discontinuity. These are revealed by finding the values of \(x\) where the denominator is zero. In our example, solving the equation \(x^{2}-5x+4=0\) will help identify points of discontinuity. When the denominator equals zero, the expression becomes undefined, causing the graph to exhibit vertical asymptotes at these points.
Understanding rational functions is crucial because they appear frequently in both pure and applied mathematics. They model various phenomena and understanding their behavior, including discontinuities, is essential for interpreting these models.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form \(ax^2 + bx + c = 0\). The solution to these equations provides values of \(x\) that make the equation true. These solutions, or roots, can be found using various methods like factoring, completing the square, or the quadratic formula.
For the rational function \(k(x)\), you encountered the quadratic equation \(x^2 - 5x + 4 = 0\). This particular equation can be factored as \((x - 1)(x - 4) = 0\) which gives us the roots \(x = 1\) and \(x = 4\). These roots are critical since they show where the function is not defined. Effectively, the roots correspond to x-values that make the denominator zero, resulting in a discontinuity.
Understanding how to solve quadratic equations is foundational for analyzing functions, especially rational functions, where knowing where the function is undefined is essential for graphing and interpreting its behavior.

Graphs of Functions

When graphing functions, especially rational ones, identifying points of discontinuity is crucial for an accurate graph representation. Discontinuities can manifest as vertical asymptotes or holes on the graph.
In the function \(k(x) = \frac{x-4}{x^{2}-5x+4}\), graphing this function using a graphing utility helps visualize where the discontinuities occur. At \(x = 1\) and \(x = 4\), the graph will show vertical asymptotes. This is because the function approaches infinity at these points, dividing by zero as the denominator becomes zero. The graph fails to touch or cross these x-values, highlighting the discontinuity.
Graphs of functions not only aid in finding these discontinuities but also help us understand the rest of the function's behavior, such as end behavior, intercepts, and the general shape of the function. By examining the graph, students can see tangible evidence of abstract mathematical principles, like discontinuities, in action.