Problem 75 - 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. $$y=x-\frac{2}{x+1}$$

Step-by-Step Solution

Answer

Based on using a graphing utility and solving the equation with y equal to zero, the x-intercepts of the function \(y=x-\frac{2}{x+1}\) can be correctly identified and verified.

Step 1: Graphing the Function

Firstly, the equation \(y=x-\frac{2}{x+1}\) should be inputted into a graphing utility in order to observe the graph and ascertain the x-intercepts.

Step 2: Identifying the X-Intercepts

From the graph, note down the points where the curve crosses the x-axis. These points at which y=0 are the x-intercepts.

Step 3: Verifying the X-intercepts

Set y equal to zero in the equation \(y=x-\frac{2}{x+1}\) and solve for x. The result should match the points identified in step 2.

Key Concepts

X-Intercepts

Identifying the x-intercepts of a function is a fundamental aspect of graphing rational functions. An x-intercept is a point on the graph where the curve crosses the x-axis, which corresponds to a location where the function's output, or y-value, equals zero. To find the x-intercepts algebraically, we set y to zero and solve the equation for x. In the given rational function y = x - \(\frac{2}{x+1}\), we would start by equating y to zero and simplifying the equation to isolate x. This process often involves factoring, multiplying through by common denominators, or using other algebraic methods to solve for x. However, when the equation can't be easily factored or is too complex, graphing utilities become very helpful in identifying these intercepts visually.

Graphing Utility

A graphing utility is a tool that allows students to visualize equations by automatically plotting the corresponding graphs. This is particularly useful when dealing with complicated functions, such as rational functions. By inputting the equation y = x - \(\frac{2}{x+1}\) into a graphing utility, students can obtain a visual representation of the function, making it easier to spot features like x-intercepts and asymptotes. The precise location of x-intercepts, as depicted on the graph, aids in confirming the results obtained through algebraic methods. Furthermore, graphing utilities can reveal the behavior of the function around these intercepts. For example, whether the graph of the function cuts through the x-axis or just touches it at the intercepts can be observed clearly through such tools.

Rational Equations

A rational equation is an equality involving at least one rational expression, which is a ratio of two polynomials. The given function y = x - \(\frac{2}{x+1}\) is an example of a rational equation with a rational expression \(\frac{2}{x+1}\). These equations often exhibit interesting characteristics, such as asymptotes and discontinuities. When graphing rational functions, it is important to recognize these features, in addition to the x-intercepts, to fully understand the behavior of the function. Solving rational equations typically entails finding a common denominator, simplifying the resulting equation, and ensuring that the solutions do not cause a zero denominator, which would be undefined.

Solving Equations

The process of solving equations is foundational to graphing rational functions. From simple linear equations to more complex rational equations, the goal is to isolate the variable of interest on one side of the equation. In the context of locating x-intercepts, setting y to zero and solving for x is a straightforward method. When faced with a rational equation, such as y = x - \(\frac{2}{x+1}\) where y is zero, it may be necessary to perform additional steps like multiplying each term by the common denominator to eliminate fractions. Ensuring that both sides of the equation balance during the manipulation is crucial, and any extraneous solutions that arise, which don't satisfy the original equation, must be excluded from the final answer.

• Set y to zero
• Isolate x on one side of the equation
• Perform algebraic operations carefully
• Check for extraneous solutions