Problem 5 - Applied Calculus

Question

For each function: a. Evaluate the given expression. b. Find the domain of the function. c. Find the range. [Hint: Use a graphing calculator. You may have to ignore some false lines on the graph. Graphing in "dot mode" will also eliminate false lines. $$ f(x)=\frac{x^{2}}{x-1} ; \text { find } f(-1) $$

Step-by-Step Solution

Answer

a. \( f(-1) = -\frac{1}{2} \); b. Domain: \( x \neq 1 \); c. Range: \( y \neq 0 \) and \( y \neq 1 \).

Step 1: Evaluate the Expression at x = -1

Substitute \( x = -1 \) into the function \( f(x) = \frac{x^2}{x-1} \). This gives you \( f(-1) = \frac{(-1)^2}{-1-1} \). Calculate the numerator and the denominator separately: \( (-1)^2 = 1 \) and \( -1-1 = -2 \). Therefore, \( f(-1) = \frac{1}{-2} = -\frac{1}{2} \).

Step 2: Determine the Domain of the Function

The domain of the function \( f(x) = \frac{x^2}{x-1} \) is all real numbers, except where the denominator is zero (since division by zero is undefined). Set the denominator equal to zero: \( x - 1 = 0 \), which gives \( x = 1 \). Thus, the domain is all real numbers except \( x = 1 \).

Step 3: Determine the Range of the Function

To find the range of \( f(x) = \frac{x^2}{x-1} \), consider its behavior and output values. The function exhibits a vertical asymptote at \( x=1 \) and a horizontal asymptote at \( y=x+1 \) as \( x \to \infty \) and \( x \to -\infty \), meaning \( y \) approaches but never equals \( x+1 \). The function can output any real number except for values between 0 and 1. Therefore, the range is all real numbers \( y eq 0 \) and \( y eq 1 \).

Key Concepts

Function Evaluation

Function evaluation refers to the process of finding the value of a function at a specific point. In our example, we have the function \( f(x) = \frac{x^2}{x-1} \), and we're tasked with evaluating it at \( x = -1 \).

To do this, we substitute \( x = -1 \) into the function. This gives:

• Numerator: \((-1)^2 = 1\)
• Denominator: \(-1-1 = -2\)

Consequently, \( f(-1) = \frac{1}{-2} = -\frac{1}{2} \). This means that when \( x = -1 \), the function outputs \( -\frac{1}{2} \). Understanding function evaluation is crucial because it allows one to decipher different values a function can assume. It serves as a foundation for analyzing a function's behavior for different input values.

Vertical Asymptote

A vertical asymptote is a vertical line that a graph of a function approaches but never crosses. It signals the values for which the function becomes undefined. In our function \( f(x) = \frac{x^2}{x-1} \), the vertical asymptote occurs where the denominator of the function is zero.

To find this, we set the denominator equal to zero:

• \( x - 1 = 0 \)
• Therefore, \( x = 1 \)

This indicates a vertical asymptote at \( x = 1 \). As \( x \) approaches 1, the function value approaches infinity or negative infinity, creating a gap in the graph. Understanding vertical asymptotes helps in graphing functions accurately and determining their domains.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of the function approaches as \( x \) moves towards positive or negative infinity. Horizontal asymptotes illustrate the behavior of a function as the inputs become very large or very small.

For the function \( f(x) = \frac{x^2}{x-1} \), we investigate its behavior as \( x \to \infty \) and \( x \to -\infty \). As we examine the function's form, the horizontal asymptote is identified as \( y = x+1 \). This indicates that as \( x \) becomes very large (in either direction), the output \( y \) gets very close to, but never quite reaches, \( x+1 \). Recognizing horizontal asymptotes is essential for understanding the long-term trends of a function's graph and its range.

Graphing Calculator Usage

A graphing calculator is a valuable tool for visualizing and analyzing functions. In this exercise, a graphing calculator helps in identifying false lines and verifying the vertical and horizontal asymptotes. It can also assist in evaluating functions and finding domains and ranges.

When using a graphing calculator, it's essential to:

• Enter the function correctly: Ensure you input the function precisely as given.
  
• Use 'dot mode' to avoid false lines: This feature eliminates erroneous connections that might appear due to the calculator attempting to draw complete lines across undefined points.
• Zoom and adjust the view: Adjust the window settings to get the best view of the function, especially near asymptotes.

Using a graphing calculator effectively can significantly enhance comprehension by providing a dynamic and visual representation of mathematical concepts.