Question

In Exercises \(21-30\) , determine whether the function is even, odd, or neither. Try to answer without writing anything (except the answer). $$y=\sqrt[3]{2-x}$$

Short answer

The function \(y = \sqrt[3]{2-x}\) is neither an even function nor an odd function.

Step-by-step solution

Propose the function as an Even function

An even function is such that \(f(-x) = f(x)\). For the given function \(y = \sqrt[3]{2-x}\), let us replace \(x\) with \(-x\) to verify this. So, we get, \(f(-x) = \sqrt[3]{2-(-x)} = \sqrt[3]{2+x}\). Since, \(f(-x)\) does not equal \(f(x)\), the given function is not an even function.

Propose the function as an Odd function

An odd function is such that \(f(-x) = -f(x)\). So, we need to check if \(f(-x)\) equals \(-f(x)\) for the given function \(y = \sqrt[3]{2-x}\). Let's replace \(x\) with \(-x\) in this function and verify this. We'll get \(f(-x) = \sqrt[3]{2+x}\). While \(-f(x) = -\sqrt[3]{2-x}\). As \(f(-x)\) does not equal \(-f(x)\), the given function is not an odd function.

Conclude the nature of function

As the function \(y = \sqrt[3]{2-x}\) does not satisfy either the characteristics of even function or odd function, we conclude that this function is neither even nor odd.

Key concepts

Symmetry of Functions

Understanding the symmetry of functions is crucial for identifying the nature of a function, particularly whether it is even, odd, or neither. Symmetry refers to the balance and proportion of a function's graph when reflected over an axis or rotated around a point.

For a function to be even, its graph should be symmetrical along the y-axis. This means that for every point \( (x, y) \) on the graph, the point \( (-x, y) \) must also be on the graph. The algebraic test for evenness is that \( f(-x) = f(x) \). However, if you find that \( f(-x) \) is not equal to \( f(x) \), as in the exercise with \( f(-x) = \sqrt[3]{2+x} \), the function fails to be even.

Conversely, an odd function shows symmetry about the origin, which means for every point \( (x, y) \) on the function's graph, the point \( (-x, -y) \) must also be present. The condition \( f(-x) = -f(x) \) must be satisfied for a function to be categorized as odd. When \( f(-x) \) does not equal \( -f(x) \) as with our function \( f(x) = \sqrt[3]{2-x} \) where \( f(-x) \) and \( -f(x) \) do not match, it can't be odd either.

Analyze the symmetry of functions to interpret their characteristics better and to understand the geometry of their graphs. This awareness can greatly inform the solution of many algebraic and calculus problems.

Algebraic Properties

When dealing with functions, algebraic properties are essential tools that allow us to manipulate and understand these functions more deeply. These properties include the familiar operations of addition, multiplication, as well as the rules for dealing with exponents and roots.

For the exercise in question, recognizing that the cube root function can handle negative inputs is important. However, the algebraic manipulation of replacing \( x \) with \( -x \) in \( f(x) = \sqrt[3]{2-x} \) to test for evenness or oddness doesn't yield an expression that matches either \( f(x) \) or \( -f(x) \) exactly; thus, we determine the function's nature based on these algebraic properties.

Furthermore, the algebraic approach to evaluating even or odd nature of functions must consider the operation within the function itself. Some functions may have complicating elements—like subtraction in our exercise—that don’t have straightforward symmetry, affecting their classification.

Understanding these algebraic properties and how they influence the symmetry and behavior of functions under transformation is key for solving and graphing complex functions.

Function Transformation

Function transformation is a process that takes a function and systematically alters its graph's appearance. This can include shifting, stretching, compressing, and reflecting the graph in various ways. Each transformation changes the function's equation in a specific manner.

However, with regards to symmetry and determining if a function is even or odd, we focus more on reflections. Reflections across the y-axis will affect how we perceive the function to be even if the original function was even. Similarly, reflections about the origin are related to functions being odd.

In our case, \( f(x) = \sqrt[3]{2-x} \) is not transformed by simple reflection, rotation, or shifting to match itself. Hence, it is classified as neither even nor odd. Recognizing the effects of function transformations on the even or odd nature of a function helps in visualizing the function’s graph and predicting its behavior, which are valuable skills in calculus and higher mathematics.