Problem 55 - Calculus of a Single Variable

Question

Comparing Functions In Exercises 55 and \(56,\) use symmetry, extrema, and zeros to sketch the graph of \(f .\) How do the functions \(f\) and \(g\) differ? $$ \begin{array}{l}{f(x)=\frac{x^{5}-4 x^{3}+3 x}{x^{2}-1}} \\\ {g(x)=x\left(x^{2}-3\right)}\end{array} $$

Step-by-Step Solution

Answer

After evaluating each function individually in terms of symmetry, extrema, and zeros and graphing these points, it is possible to observe and list the differences between the two functions based on their graphs.

Step 1: Evaluate Symmetry, Extrema, and Zeros of Function \(f(x)\)

First, determine whether function \(f(x)\) is even, odd, or neither by substituting \(-x\) for \(x\) and simplifying. After this, locate the extrema by finding the derivative, setting it equal to zero, and finding the \(x\)-values at these points. Determine whether these points are local maxima or minima using the second derivative. Find the zeros by setting the function equal to zero and solving for \(x\). Plot these points on a graph.

Step 2: Evaluate Symmetry, Extrema, and Zeros of Function \(g(x)\)

Repeat the above process for function \(g(x)\). Determine symmetry, locate extrema, find zeros, and plot these points on a separate graph.

Step 3: Compare the two functions

Compare the two functions' graphs and state the differences. Pay attention to how the extrema, symmetry, and zeros differ between the two functions.

Key Concepts

Symmetry in Functions

Symmetry in functions is an important concept that helps with graphing by revealing patterns or behaviors that mirror each other. There are two main types of symmetry:

• Even Functions: A function is even if it is symmetric with respect to the y-axis. This means that for every point \((x, y)\) on the graph, there is a corresponding point \((-x, y)\). Mathematically, if a function is even, then \(f(x) = f(-x)\) for all \(x\).
• Odd Functions: A function is odd if it is symmetric with respect to the origin. This implies that for every point \((x, y)\), there is a point \((-x, -y)\). If a function is odd, then \(f(-x) = -f(x)\).

To determine the symmetry of a function \(f(x)\), substitute \(-x\) in place of \(x\) in the equation and simplify. If it results in the original function, it is even; if it results in the negative of the original, it is odd. For the function \(f(x)=\frac{x^{5}-4 x^{3}+3 x}{x^{2}-1}\), you substitute and simplify to identify symmetry. Recognizing symmetry can simplify graphing since only half of the equation may be needed to plot.

Finding Extrema

Finding extrema, which include maxima (highest points) and minima (lowest points), is crucial for understanding the behavior of a function's graph. Extrema can be classified as local (relative) or absolute (global).

• Local Extrema: A function \(f(x)\) has a local maximum at a point if \(f(x)\) is higher than at any nearby points. Similarly, it has a local minimum if \(f(x)\) is lower than at any nearby points.
• Absolute Extrema: This is the highest or lowest point on the entire graph of the function.

To find these points, you need to first calculate the derivative of the function, \(f'(x)\). Set \(f'(x) = 0\) and solve for \(x\) to find critical points. These are potential extrema. Use the second derivative test to determine the nature of these points:

• If \(f''(x) > 0\) at a critical point, there's a local minimum.
• If \(f''(x) < 0\), it's a local maximum.
• If \(f''(x) = 0\), further tests are needed.

For \(f(x)=\frac{x^{5}-4 x^{3}+3 x}{x^{2}-1}\), these steps will help sketch the graph by pinpointing where it rises or falls.

Zeros of Functions

Zeros of a function are the x-values where the function equals zero. These points, also known as roots or x-intercepts, are crucial in graphing as they identify where the graph crosses or touches the x-axis.
To find these zeros, set \(f(x) = 0\) and solve the resulting equation for \(x\). The solutions are the x-intercepts of the function's graph. For example, for \(f(x)=\frac{x^{5}-4 x^{3}+3 x}{x^{2}-1}\), you'll set \[\frac{x^{5}-4 x^{3}+3 x}{x^{2}-1} = 0\]and solve for \(x\) by finding values that make the numerator zero while the denominator is not zero.
Knowing these zeros is essential because they divide the function into intervals which help in analyzing where the function is positive or negative. Furthermore, they substantially assist in sketching the behavior and shape of the graph. This process helps visualize intersections with axes and is fundamental in understanding any graphical representation of functions.