Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Compare the graph of \(f(x)=4 / x^{2}\) with the graph of \(g\). $$g(x)=-f(x)=-\frac{4}{x^{2}}$$

Step-by-Step Solution

Answer

The graphs of \(f(x)\) and \(g(x)\) have similar shapes and symmetries but are reflected about the x-axis. The graph of \(f(x)\) is always positive, whereas the graph of \(g(x)\) is always negative.

Step 1: Draw the graph of \(f(x)\)

The function \(f(x) = 4 / x^2\) is a reciprocal square function. So, the graph will have a characteristic shape: It is undefined at \(x = 0\), positive for every \(x\), goes to infinity as \(x\) approaches zero, and tends towards \(0\) as \(x\) goes to \(-\infty\) or \(+\infty\). Note, the graph is symmetric with respect to y-axis.

Step 2: Draw the graph of \(g(x)\)

The equation \(g(x) = -f(x) = -4 / x^2\) is merely the graph of \(f(x)\) reflected over the x-axis because multiplying a function by \(-1\) does just that. So, \(g(x)\) is also undefined at \(x = 0\), now goes to \(-\infty\) as \(x\) approaches zero, and still tends towards \(0\) as \(x\) goes to \(-\infty\) or \(+\infty\). Again, the graph is symmetric with respect to y-axis.

Step 3: Comparison of \(f(x)\) and \(g(x)\)

The graph of \(f(x)\) and \(g(x)\) are identical, however they are reflected about the x-axis. They are both undefined at \(x=0\), symmetric about y-axis and tend toward \(0\) as \(x\) goes to \(-\infty\) or \(+\infty\). The key difference is that \(f(x)\) always produce positive outputs while \(g(x)\) produce negative outputs.

Key Concepts

Reciprocal Square Function

A reciprocal square function follows the form \(f(x) = \frac{a}{x^2}\), where \(a\) is a constant. These functions exhibit specific characteristics that make them easily distinguishable when graphed. The function \(f(x) = \frac{4}{x^2}\) in our exercise is a prime example of such a function.

The graph will show that as \(x\) approaches zero, the \(y\)-values of \(f(x)\) increase without bound, that is, the function goes to infinity. This behavior represents vertical asymptotes, lines that the graph approaches but never touches or crosses. Moreover, because the denominator is squared, the function never takes negative values, so the graph is located entirely in the upper half of the coordinate plane. This makes the graph symmetric about the \(y\)-axis. As \(x\) moves towards negative or positive infinity, the value of \(f(x)\) approaches zero, indicating horizontal asymptotes. When graphing such functions, it is important to plot points for different values of \(x\) and to draw smooth curves that approach the asymptotes.

Reflection Over X-axis

Reflection over the \(x\)-axis occurs when each point of the original function is flipped across the \(x\)-axis, essentially changing the sign of the \(y\)-coordinate while leaving the \(x\)-coordinate unchanged. For a function \(f(x)\), the reflection over the \(x\)-axis is represented by \(g(x) = -f(x)\). This transformation results in a graph that is a mirror image of the original graph across the \(x\)-axis.

In our exercise, \(g(x) = -\frac{4}{x^2}\) represents the reflection of \(f(x)\). It's crucial to note that points on the \(x\)-axis remain unchanged since their \(y\)-coordinates are zero. When performing a reflection over the \(x\)-axis, it's helpful to plot some corresponding points of \(f(x)\) and their reflected points for \(g(x)\), and then connect these points with a smooth curve, being mindful of the behavior near asymptotes.

Asymptotes in Functions

Asymptotes are lines that a graph of a function approaches but never actually touches. They can be vertical, horizontal, or even oblique. In the context of reciprocal functions like \(f(x) = \frac{4}{x^2}\), there are two key types of asymptotes we encounter:

• Vertical Asymptotes: These occur at values of \(x\) that cause the function to be undefined, typically where the denominator equals zero. In our exercise, \(x = 0\) yields a vertical asymptote.
• Horizontal Asymptotes: These are horizontal lines that the graph approaches as \(x\) goes to infinity or negative infinity. For \(f(x)\), the horizontal asymptote is the \(x\)-axis, or \(y=0\), since the function's values get closer and closer to zero the further away \(x\) moves from the origin.

Understanding asymptotes is vital for accurately sketching the behavior of functions, especially near the edges of the graph. Asymptotes serve as guides that constrain the extent of the graph and communicate the function's behavior at extreme values of \(x\).