Problem 72 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=\frac{1}{x-2}-\frac{5}{x}$$

Step-by-Step Solution

Answer

The x-intercepts of the function \(y=\frac{1}{x-2}-\frac{5}{x}\) are \(x=2\) and \(x=5\).

Step 1: Graph the Function

The function \(y=\frac{1}{x-2}-\frac{5}{x}\) is graphed using a graphing utility. The x-intercepts are determined from the graph.

Step 2: Set \(y=0\) to Confirm

The function is set to zero and simplified, \(0=\frac{1}{x-2}-\frac{5}{x}\), then multiplied through by \(x(x-2)\) to clear the fractions:\(0=x(x-2)-5(x-2)\), then distributive property is applied:\(0=x^2-2x-5x+10\), then combine like terms:\(0=x^2-7x+10\).

Step 3: Solve for \(x\)

The resulting quadratic equation is then factored and solved for \(x\):\((x-2)(x-5)=0\), which gives the solutions \(x=2\) and \(x=5\).

Key Concepts

X-Intercepts

The concept of x-intercepts of a graph of a function is fundamental to understanding how the function behaves. X-intercepts are the points where the graph of the function crosses the x-axis. This means the output or the y-value is zero at these points. For rational functions, like the given function \(y=\frac{1}{x-2}-\frac{5}{x}\), finding the x-intercepts requires setting \(y=0\) and solving the equation.

Let's take a closer look at the provided problem. After setting \(y=0\) and clearing the fractions, you're left with a quadratic equation, which is the main type of equation where x-intercepts come into play. An important aspect to remember here is that x-intercepts can offer insights into the potential symmetry of a graph and can help in sketching its rough shape even before using any graphing utilities.

Graphing Utility Usage

Graphing utilities are powerful tools used for visualizing functions and their properties. When you graph the rational function \(y=\frac{1}{x-2}-\frac{5}{x}\), you utilize technology to see how it behaves across different values of x. This visual representation makes it easier to spot key features like x-intercepts, asymptotes, and intervals of increase or decrease.

When using a graphing utility, especially for complex functions such as rational ones, it's important to adjust the window settings to ensure all relevant parts of the graph are visible. This helps in accurately identifying where the graph crosses the x-axis. Additionally, modern graphing utilities often offer features that automatically calculate and display x-intercepts, which can be verified through algebraic methods as shown in the textbook solution.

Solving Quadratic Equations

Solving quadratic equations is a vital skill in mathematics. These equations often appear when dealing with parabolas and can be recognized by the highest power of x being 2, hence the name 'quadratic'. The general form is \(ax^2+bx+c=0\).

To solve them, you can use factoring, completing the square, or the quadratic formula. In the case of our exercise, factoring is used to find the values of x that make the equation true - those are the x-intercept(s) of the function. When the textbook exercise presents \((x-2)(x-5)=0\), it shows that \(x=2\) and \(x=5\) are the roots of the quadratic and thus the x-intercepts of the rational function. Remember that not all quadratics will factor nicely, and alternative methods may be required. It's always good to verify the solutions graphically, as it provides a clear confirmation of the intercepts.