Problem 10 - Calculus of a Single Variable

Question

In Exercises 5–24, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$ f(x)=\frac{x-3}{x} $$

Step-by-Step Solution

Answer

The graph of function \(f(x)=\frac{x-3}{x}\) has an x-intercept at \(x=3\), no y-intercept, a vertical asymptote at \(x=0\), and a horizontal asymptote at \(y=1\). There are no relative minima, maxima, or inflection points. The function decreases from \(- \infty\) to \(0\) and increases from \(0\) to \(\infty\).

Step 1: Find X and Y Intercepts

To find the x-intercept, set \(f(x) = 0\), which gives \(x - 3 = 0\), meaning \(x = 3\) is the x-intercept. To find the y-intercept, set \(x = 0\), but unfortunately this makes the function undefined, hence there is no y-intercept.

Step 2: Investigate Asymptotes

To determine the asymptotes, for vertical asymptote evaluate the value of \(x\) that makes the denominator equal to zero, the function is undefined at \(x = 0\), hence \(x = 0\) is a vertical asymptote. As for horizontal asymptotes, as \(x \to \pm \infty\), \(f(x) \to 1\), hence \(y = 1\) is the horizontal asymptote.

Step 3: Calculate the Derivative for Extrema and Inflection Points

To find extrema, derive the function and set it to zero, \(f'(x) = -\frac{3}{x^2}\), which yields no critical points since \(-\frac{3}{x^2} = 0\) has no real solutions. The function's second derivative for inflection points is \(f''(x) = \frac{6}{x^3}\), setting this to zero also gives no results. So, there are no relative extrema and inflection points.

Step 4: Graphing the Function

Plot the function on a graph, indicating the x-intercept at \(x=3\), the vertical asymptote at \(x=0\) and horizontal asymptote at \(y=1\). The function is decreasing on the interval \((- \infty, 0)\) and increasing on the interval \((0, \infty)\). You can use a graphing utility to verify this.

Key Concepts

X-Intercept and Y-Intercept

The x-intercept of a graph is the point where the function crosses the x-axis. For the function \( f(x)=\frac{x-3}{x} \), finding the x-intercept involves setting \( f(x) \) equal to zero. Solving \( 0=\frac{x-3}{x} \) reveals that the x-intercept occurs at \( x=3 \). This point can be represented on a graph as (3, 0).

On the other hand, the y-intercept is where the graph crosses the y-axis. This is found by evaluating the function at \( x=0 \). However, for our function \( f(x) \), setting \( x=0 \) makes the function undefined, meaning the graph does not intersect the y-axis at any point. This distinction is crucial for understanding the behavior of rational functions, as y-intercepts do not always exist, especially when the y-axis is a vertical asymptote.

Vertical and Horizontal Asymptotes

Asymptotes are lines that the graph of a function approaches but never actually touches. Vertical asymptotes occur at values of \( x \) where the function becomes undefined. For the given function \( f(x)=\frac{x-3}{x} \), the denominator equals zero when \( x=0 \), thus the line \( x=0 \) is a vertical asymptote of the graph.

Horizontal asymptotes happen when the graph of the function approaches a particular y-value as \( x \) goes towards infinity. For \( f(x) \), as \( x \) approaches either positive or negative infinity, the function simplifies to \( y=1 \), indicating a horizontal asymptote. Identifying these asymptotes provides a skeletal structure to which the curve of the function adheres, and is essential for sketching an accurate graph.

Derivative for Extrema and Inflection Points

Determining the extrema (maximums and minimums) and inflection points of a function involves calculus, specifically the use of derivatives. The first derivative of a function, \( f'(x) \), can be used to identify extrema by setting it equal to zero and solving for \( x \). However, for \( f(x)=\frac{x-3}{x} \), the first derivative \( f'(x) = -\frac{3}{x^2} \) never equals zero and thus indicates no extrema are present.

Inflection points, locations where the curve changes concavity, are found using the second derivative. For our function, \( f''(x) = \frac{6}{x^3} \) also does not equal zero for any real \( x \), signaling no inflection points. Understanding where these features occur is key to identifying the shape and turning points of the graph.

Graphing Utility Verification

A graphing utility is a tool or software used to visually render mathematical functions. After performing a manual analysis of a function, it's beneficial to verify key features like intercepts and asymptotes using such a utility. When graphing \( f(x)=\frac{x-3}{x} \), ensure that it includes the x-intercept at \( x=3 \), the vertical asymptote at \( x=0 \), and the horizontal asymptote at \( y=1 \). The utility can vividly demonstrate how the function behaves and offer a visual confirmation of the function's decreasing trend on \( (- \infty, 0) \) and increasing trend on \( (0, \infty) \). Verification through a graphing utility not only supports the accuracy of the manual graph but can also be crucial for catching potential errors in the analytical process.