Problem 61 - Essential Calculus

Question

Use symmetry, extrema, and zeros to sketch the graph of \(f .\) How do the functions \(f\) and \(g\) differ? Explain. $$ f(x)=\frac{x^{5}-4 x^{3}+3 x}{x^{2}-1}, \quad g(x)=x\left(x^{2}-3\right) $$

Step-by-Step Solution

Answer

The functions \(f(x)\) and \(g(x)\) differ in terms of their symmetry, zeros, and extrema, which leads to different shapes and behaviors of their graphs. To understand these differences in more detail, you'll need to work through the previously mentioned steps.

Step 1: Find Symmetry of Functions

To find symmetry of the functions, substitute \(-x\) for \(x\) in both functions. If \(f(-x) = f(x)\), then the function is even and the graph of the function will be symmetric with respect to the y-axis. If \(f(-x) = -f(x)\), then it is odd and the graph of the function will be symmetric with respect to the origin.

Step 2: Find Zeros and Extrema

To find zeros of the functions, set each function equal to zero and solve for \(x\). These are the x-intercepts of the graph. To find extrema, take the derivative of the function, set equal to 0, and solve for \(x\). These are potential maxima or minima points.

Step 3: Sketch the Graphs

Based on the gathered information about symmetry, zeros, and extrema, sketch a graph for each function. Use the zeros for x-intercepts and use the extrema for potential high and low points in the graph.

Step 4: Comparing Functions

Compare the characteristics (symmetry, zeros, extrema) and the graphs of the two functions to point out how the functions \(f(x)\) and \(g(x)\) differ.

Step 5: Explain the Differences

Explain the differences between the functions based on the characteristics and sketch of the graphs. This might include differences in symmetry, number and location of zeros and extrema, and the overall shape and behavior of the graph.

Key Concepts

Function Symmetry

When we talk about symmetry in functions, we're exploring how the graph of a function relates to itself under specific transformations. To determine symmetry, you substitute \(-x\) for \(x\) in the function.

- **Even Functions:** If \(-x\) results in the original function (\(f(-x) = f(x)\)), it’s symmetric with respect to the y-axis. This means that the left and right sides of the graph mirror each other.- **Odd Functions:** If substituting \(-x\) results in the negative of the original function (\(f(-x) = -f(x)\)), it’s symmetric with respect to the origin. Here the graph is rotationally symmetrical around the origin.

Understanding symmetry helps in predicting the shape of the graph and the behavior of the function without plotting too many points. This simplifies the sketching process and provides insight into the function’s behavior on both sides of the y-axis.

Zeros of Functions

The zeros of a function are the points where the function crosses the x-axis. These are solutions to the equation \(f(x) = 0\). Finding zeros is crucial because:

• They show where the graph touches or crosses the x-axis.
• Provide critical insights into the solution of equations related to the function.

To find zeros:- Set the function equal to zero.- Solve for \(x\) to find the x-intercepts.

For rational functions like \(rac{x^{5}-4x^{3}+3x}{x^{2}-1}\), remember to consider simplifying the numerator and denominator separately. Also, ensure the denominator isn’t zero since that will create undefined points rather than zeros. Zeros essentially map out critical points that define where a graph starts or ends, or how it behaves in different intervals.

Extrema of Functions

Extrema are points on the function where a local maximum or minimum occurs. These are the peaks or valleys in the graph and can help identify where the function changes direction.

To find these points:

• Take the derivative of the function.
• Set the derivative equal to zero and solve for \(x\) to find critical points.

These points are potential candidates for maxima or minima. After identifying the critical points, use the second derivative test or evaluate nearby values to confirm the type of extremum for each point.

Recognizing extrema allows us to understand the overall shape and flow of the function. It pinpoints where significant changes in direction occur, thus playing a crucial role in graph sketching and analysis. The behavior of the function at these extrema can provide insights into real-world phenomena modeled by the function.