Problem 79 - 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. $$y=x+1+\frac{2}{x-1}$$

Step-by-Step Solution

Answer

The graphical representation will show the x-intercepts of the given function. Solving the equation algebraically by setting \(y\) to 0, simplifying and solving for \(x\) will confirm these intercepts. The main points are plugging the obtained \(x\) values back into the equation to verify if they satisfy the equation.

Step 1: Graph the function

Using a graphing utility, plot the function \(y = x + 1 + \frac{2}{x-1}\). Here, observe the graph to identify where it crosses the x-axis. These crossing points are the x-intercepts.

Step 2: Set y to 0

In order to solve and confirm the x-intercepts algebraically, set \(y\) to 0 in the equation: \[0 = x + 1 + \frac{2}{x-1},\] then isolate \(x\).

Step 3: Clear the fraction

Multiply every term by \(x-1\) to clear the fraction: \[0=(x+1)(x-1)+2\].

Step 4: Simplify the equation

Expand and simplify the equation to its simplest form, then solve the quadratic equation to get \(x\)'s values.

Step 5: Confirm your result

Check the x-values obtained by substituting them back into the equation. Your solutions are correct if the left-hand side equals the right-hand side.

Key Concepts

X-intercepts

The x-intercepts of a function are the points where the graph of the function crosses the x-axis. These are the values of the variable x when the output, or the y-value, is zero. To calculate the x-intercepts of a function algebraically, one must set the y-value to zero and solve for x.

For instance, consider the function provided in the exercise: \(y = x + 1 + \frac{2}{x-1}\). To find its x-intercept(s), we begin by setting \(y = 0\), giving us the equation \(0 = x + 1 + \frac{2}{x-1}\). This equation can be solved to find the x-values at which the graph intersects the x-axis. It's important for learners to both understand and visualize this concept. The graph provides an immediate visual cue to the x-intercepts, supplementing the mathematical approach and confirming the accuracy of the algebraic solution.

Graphing Utilities

Graphing utilities are technological tools that help to construct the visual representation of functions. They can be found in the form of software programs, apps, or online graphing calculators. Many students benefit from using these utilities as they provide an immediate visual representation of complex functions.

When working with the given function \(y = x + 1 + \frac{2}{x-1}\), the graphing utility helps by swiftly showing where the graph crosses the x-axis, revealing the x-intercepts without manual plotting. This visual aid makes it simpler for students to comprehend the function's behavior, especially near the intercepts. When combined with the step of solving algebraically, graphing tools ensure a comprehensive understanding by correlating the visual graph with the algebraic solution.

Solving Algebraic Equations

Solving algebraic equations is a fundamental skill in mathematics. It involves finding the value(s) of the variable(s) that make the equation true. Steps often include simplifying the expression, isolating the variable, and performing arithmetic operations to solve for the variable.

To solve the equation from the provided exercise, \(0 = x + 1 + \frac{2}{x-1}\), we begin by eliminating the fraction to simplify the equation—a key step in solving algebraic fractions. Multiplying every term by \(x-1\), we obtain a quadratic equation, which can then be expanded, simplified, and solved. The solutions of this quadratic are the x-values that correspond to the x-intercepts on the graph. This process demonstrates the interconnection between graphical solutions and algebraic manipulations in finding the x-intercepts of a function.