Question

In Problems 37-48, determine algebraically whether each function is even, odd, or neither. \(f(x)=4 x^{3}\)

Short answer

The function \[ f(x)=4 x^{3} \] is odd.

Step-by-step solution

- Understanding Even and Odd Functions

A function is considered even if for every x in the function's domain, the equality \[ f(-x) = f(x) \] holds true. A function is considered odd if \[ f(-x) = -f(x) \] for all x in the function’s domain.

- Find f(-x)

Substitute \[ -x \] into the given function. Start by writing the original function: \[ f(x) = 4x^3 \]. Now, find \[ f(-x) \]: \[ f(-x) = 4(-x)^3 \].

- Simplify f(-x)

Simplify the expression \[ 4(-x)^3 \]: \[ f(-x) = 4(-x)^3 = 4(-x^3) = -4x^3 \].

- Compare f(-x) and -f(x)

Recall the original function \[ f(x) = 4x^3 \]. Find \[ -f(x) \]: \[ -f(x) = -(4x^3) = -4x^3 \].

- Determine if the Function is Odd

Compare \[ f(-x) \] and \[ -f(x) \]. Since \[ f(-x) = -4x^3 \] and \[ -f(x) = -4x^3 \], we see that \[ f(-x) = -f(x) \]. Therefore, the function is odd.

Key concepts

function symmetry

Function symmetry is an important concept in mathematics. It tells us about the visual properties of a function when plotted on a graph. Symmetry can help us easily identify whether a function is even, odd, or neither. For even functions, the graph is symmetric about the y-axis. This means if you fold the graph along the y-axis, both halves would line up perfectly. For odd functions, the graph is symmetric about the origin. This means that rotating the graph 180 degrees around the origin will leave the graph unchanged. Symmetry is not just visually appealing but also has algebraic implications and practical applications in various fields.

algebraic functions

Algebraic functions are those that involve variables, constants, and the basic operations: addition, subtraction, multiplication, division, and raising to a power. These functions can have either simple expressions like linear functions or more complex ones like polynomial functions. Understanding algebraic functions is crucial as they form the basis of more advanced mathematical concepts. In the context of even and odd functions, algebraic manipulation helps us determine their nature. For example, substituting \(x\) with \(-x\) in a given function is an algebraic method used to verify function symmetry.

polynomial functions

Polynomial functions are a type of algebraic function that involve sums of powers of variables with coefficients. The general form of a polynomial function is \ f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \ , where \ a_i \ are constants. One of the key properties of polynomial functions is that their degree (the highest power of the variable) influences their shape and symmetry. For instance, in our exercise, \[ f(x) = 4x^3 \] is a cubic polynomial function which is odd because substituting \[ -x \] resulted in \[-4x^3\], making \[-f(x) = -4x^3 \]. Thus, polynomial functions like this one help in studying not just algebraic solutions but also their graphical behaviours.

function properties

Function properties encompass the characteristics and behaviours that define different types of functions. These include domain and range, continuity, limits, symmetry, periodicity, and more. Understanding these properties is essential for analyzing and interpreting functions correctly. Even and odd properties are among these critical attributes which help in predicting the behavior of functions under specific transformations. In our case, the property indicating that \[ f(x) = 4x^3 \] is odd shows that it will have symmetry around the origin. Knowing these properties aids in solving and simplifying complex mathematical problems, making it easier for students to grasp advanced concepts.