Problem 77 - 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. $$y=x+2-\frac{1}{x+1}$$

Step-by-Step Solution

Answer

The x-intercepts are the solutions of the equation set for \(y=0\). Those points can be confirmed by solving \(0 = x + 2 - \frac{1}{x + 1}\).

Step 1: Use a Graphing Utility

Input the function \(y = x + 2 - \frac{1}{x + 1}\) into a graphing utility. The shape of the graph will give a visual representation of how the function behaves.

Step 2: Determine x-intercepts

Identify the points (if any) where the function crosses or touches the x-axis. These are the x-intercepts of the function where \(y = 0\). Remember, there may be none, one, or more than one x-intercept.

Step 3: Confirm result

Substitute \(y=0\) in the equation \(0 = x + 2 - \frac{1}{x + 1}\) and solve for x. This confirms if the identified x-intercepts on the graph are correct. To solve the equation, first simplify it by multiplying all terms by \(x+1\) to eliminate the denominator, and then solve for x.

Key Concepts

X-Intercepts

X-intercepts are crucial points on a graph where the function crosses the x-axis. This happens when the output value of the function, or \( y \), is zero. To find these intercepts visually, we can use graphing utilities that display the graph's behavior. Spotting where the graph touches or crosses the x-axis gives a clear indication of the x-intercepts.

In the given exercise, after graphing the function \( y = x + 2 - \frac{1}{x + 1} \), you'll need to carefully inspect the graph for points where the curve meets the x-axis. These locations are where \( y \) equals zero, marking potential x-intercepts. Once identified on the graph, the next step is to confirm these points analytically by solving the function equation algebraically.

Solving Equations

Solving equations involves finding the values of the variable that make the equation true. For the function in our exercise, \( y = x + 2 - \frac{1}{x + 1} \), we set \( y = 0 \) to find the x-intercepts. The equation becomes \( 0 = x + 2 - \frac{1}{x + 1} \).

To solve this equation, one effective method is to eliminate the fraction. Multiply through by \( x + 1 \), the denominator, to simplify the equation:

• This gives \( 0 = (x + 1)(x + 2) - 1 \).
• Expand and rearrange terms to form a standard quadratic equation or linear form, if applicable.

Once the equation is simplified, solve for \( x \) to identify where \( y = 0 \) analytically confirms the x-intercepts observed on the graph. Each solution corresponds to an identified x-intercept, solidifying the visual analysis with mathematical verification.

These solutions help validate our interpretations of the graph and ensure our results are mathematically sound.

Using Graphing Utilities

Graphing utilities are powerful tools that visually represent functions, aiding in understanding complex mathematical concepts like intercepts and asymptotic behavior. In the context of our exercise, a graphing utility simplifies the challenge of spotting where the function \( y = x + 2 - \frac{1}{x + 1} \) crosses the x-axis, offering a visual method to identify x-intercepts.

Graphing utilities can be software or calculators that allow you to input a function and immediately view its graphical representation. They automate the graphing process, saving time and reducing errors associated with hand-drawing.

Upon inputting the function:

• Observe the overall shape of the graph.
• Look for points where the graph cuts across the x-axis; these indicate potential x-intercepts.

This visual inspection is a preliminary step before analytical confirmation, providing a quick and intuitive understanding of the function's behavior.

Rational Functions

Rational functions are functions that involve fractions where both the numerator and the denominator are polynomials. In this exercise, our function \( y = x + 2 - \frac{1}{x + 1} \) is partly rational because of the \( \frac{1}{x+1} \) term. Understanding rational functions includes recognizing their distinct characteristics, such as vertical asymptotes, which occur where their denominators are zero.

In our function, \( x + 1 \) is in the denominator, implying a vertical asymptote at \( x = -1 \), where the function is undefined. This asymptote plays a crucial role in shaping the graph and can influence the x-intercepts significantly by dividing the graph into distinct sections.

When working with rational functions, it’s important to:

• Identify and mark vertical asymptotes, remembering the function behaves differently near them.
• Understand how the behavior near these points affects intercept identification.
• Consider the overall behavior of the function, especially as it approaches asymptotes or infinity.

Recognizing these characteristics helps in accurately graphing the function and understanding its potential x-intercepts and overall behavior.