Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{1}{x-3}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\frac{1}{x-3}\) has no x-intercept, y-intercept at \((0,-\frac{1}{3})\), no symmetry, a vertical asymptote at \(x = 3\) and a horizontal asymptote at \(y = 0\).

Step 1: Identify Intercepts

For the x-intercept, you solve the equation \(f(x) = 0\). Yet, rational functions with denominator other than a constant never touch the x-axis, so there will be no x-intercept. For the y-intercept, replace \(x\) with \(0\). Thus, \(f(0)= \frac{1}{0-3}= -\frac{1}{3}\), so the y-intercept is at \((0,-\frac{1}{3})\).

Step 2: Check For Symmetry

Verify if the function is either even or odd. Remember, even functions are symmetric with respect to y-axis, and odd functions are symmetric with respect to the origin. A function \(f(x)\) is even if \(f(-x) = f(x)\) and odd if \(f(-x) = -f(x)\). When we compute \(f(-x) = \frac{1}{-x-3}\) we observe that it equals to neither \(f(x)\) nor \(-f(x)\), thus the function is neither even nor odd; there's no symmetry.

Step 3: Finding Asymptotes

Vertical asymptotes: These are the values of x such that f(x) approaches infinity. For the rational function, the vertical asymptote is when \(x = 3\) because at \(x = 3\) the function is undefined. Horizontal asymptotes: These are the values that f(x) approaches as \(x\) approaches positive or negative infinity. For this rational function, the horizontal asymptote is \(y = 0\). This is because as \(x\) gets larger or smaller, \(f(x)\) approaches zero.

Key Concepts

Graph Sketching

Sketching the graph of a rational function can seem challenging, but by breaking it down into steps, it becomes manageable. A rational function is a fraction where both the numerator and the denominator are polynomials. In the given exercise, the function is simple: \(f(x) = \frac{1}{x-3}\). This means that for each value of \(x\), you will take the reciprocal of \((x-3)\). The graph of this type of function typically involves a curve that approaches lines called asymptotes. More on that soon! The process of sketching encompasses understanding its behavior at intercepts, its symmetry, and asymptotes.

Asymptotes

Asymptotes are the invisible lines that your graph gets very close to but never touches or crosses. They guide the overall direction of the graph.

• Vertical Asymptotes: For \(f(x) = \frac{1}{x-3}\), the vertical asymptote is at \(x = 3\). This happens because the function is undefined when the denominator is zero, meaning you cannot have a zero denominator in a rational function. As \(x\) gets closer to 3 from either side, \(f(x)\) sharply increases to infinity or decreases to negative infinity.
• Horizontal Asymptotes: These are different from vertical asymptotes in that they describe the behavior of the function as \(x\) becomes infinitely large (positive or negative). For our function, the horizontal asymptote is \(y = 0\), meaning as \(x\) grows larger or smaller, \(f(x)\) approaches zero but never actually reaches it.

Recognizing these asymptotes is crucial in graph sketching as they tell you where the function cannot go.

Intercepts

Intercepts are vital points where the graph crosses the axes.

• X-Intercept: To find the x-intercept, set the numerator of the rational function equal to zero. However, in this function, since the numerator is a constant 1, and the denominator cannot be zero, the function does not cross the x-axis. Therefore, there is no x-intercept here.
• Y-Intercept: For the y-intercept, substitute \(x = 0\) into the function: \(f(0) = \frac{1}{0-3} = -\frac{1}{3}\). So, the graph will cross the y-axis at \(y = -\frac{1}{3}\). This is one of the coordinates that the function travels through, helping you visualize the graph's position relative to the axes.

Symmetry

Symmetry in a function tells us visually how the graph relates to itself around the y-axis, x-axis, or origin. This function, however, is neither symmetric about the y-axis nor about the origin. You can test for symmetry by examining the function when you substitute \(-x\) for \(x\).

• If \(f(-x) = f(x)\), the function is symmetric about the y-axis (even parity).
• If \(f(-x) = -f(x)\), it is symmetric about the origin (odd parity).

For \(f(x) = \frac{1}{x-3}\), \(f(-x) = \frac{1}{-x-3}\). Neither \(f(-x) = f(x)\) nor \(f(-x) = -f(x)\) are true, so this function lacks symmetry. This helps us know that the graph does not mirror itself across these lines.