Question

In Exercises 47-50, use a graphing calculator to graph the function. Then determine whether the function is even, odd, or neither. $$ f(x)=\frac{4 x^2}{2 x^3-x} $$

Short answer

The final answer will depend on the mathematical tests conducted in steps 2 and 3, as well as the observed symmetry in the function graph from step 1.

Step-by-step solution

Graph the Function

Enter the function \(f(x)=\frac{4 x^2}{2 x^3-x}\) into a graphing calculator and analyze the graph. Note the symmetry of the plot: if it is symmetrical about the y-axis or symmetrical about the origin.

Even Function Test

An even function, by definition, is a function in which the following rule is true: \(f(x) = f(-x)\) for all x in the function's domain. Check the given function using this rule. Substitute \(-x\) in for \(x\) in the function \(f(x)\) and simplify. If the function is simplified back to its original form, then it is even.

Odd Function Test

An odd function, by definition, is a function in which the following rule is true: \(f(-x) = -f(x)\) for all x in the function's domain. Check the given function using this rule. Substitute \(-x\) in for \(x\) in the function \(f(x)\) and simplify. If the family emerged is \(-f(x)\), then it’s odd.

Conclusion

Based on the outcomes of steps 2 and 3, determine whether the function is even, odd, or neither.

Key concepts

Algebra 2

Algebra 2 is an advanced level of mathematical study that further explores functions, polynomials, and equations, building on the foundation laid in Algebra 1. In the context of graphing rational functions, Algebra 2 students are frequently challenged to understand and analyze the behavior of functions, including observing their symmetries and determining if they are even, odd, or neither.

One important aspect of Algebra 2 is developing the ability to manipulate and simplify complex algebraic expressions. In the case of the exercise provided, simplifying the function demonstrates an understanding of polynomial division, a significant skill in Algebra 2. Students must also be fluent in recognizing the impact of negative inputs on a function's output, which is key when testing for even or odd symmetry.

Even and Odd Functions

Identifying even and odd functions is a fundamental concept in the study of symmetrical properties in mathematics. An even function is defined by the property that for every that belongs to the function's domain. It can be graphically identified by its symmetry with respect to the y-axis.

Odd functions, on the other hand, satisfy the property wherefor every that belongs to the domain. These functions are symmetrical about the origin. The exercise demonstrates a practical application of these definitions. By substituting and comparing the result to the original function or its opposite, one can determine the nature of the function's symmetry.

Graphing Calculator Usage

Graphing calculators are invaluable tools in Algebra 2, especially when dealing with complex functions. They allow students to visualize the behavior of these functions by generating their graphs swiftly. When graphing the function students can analyze the graph to observe key features such as intercepts, asymptotes, and particularly, symmetry, which relates to whether a function is even or odd.

Understanding how to input expressions properly and interpret the resulting graphs on a calculator is an essential skill. For the exercise given, using a graphing calculator simplifies the process of identifying function symmetry, which would otherwise require expansive algebraic manipulation.

Symmetry in Functions

Symmetry in functions is an insightful concept when graphing rational functions. It refers to the balance and mirroring of a graph either across the y-axis or about the origin. A symmetrical graph indicates that the function possesses either even or odd properties.

In the given exercise, recognizing symmetry serves not only to understand the behavior of but also to classify it correctly. Symmetry simplifies the determination of whether a function is even or odd because we can often make this identification visually and corroborate it algebraically. This visual assessment is a powerful tool when paired with the analytical even and odd function tests described in the previous steps.