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Chapter 1: Problem 21

In Exercises \(21-30\) , determine whether the function is even, odd, or neither. Try to answer without writing anything (except the answer). $$y=x^{4}$$

Short Answer

Expert verified
The function \(y = x^{4}\) is even.

Step by step solution

01

Understand the question

Here, it is asked to find out if the function \(y = x^{4}\) is even, odd or neither. An even function is one for which \(f(x) = f(-x)\) for every \(x\) in the function's domain. An odd function is one for which \(f(x) = -f(-x)\). If a function doesn't satisfy either of these conditions, it is neither even nor odd.
02

Substitute -x for x

We will substitute \(-x\) for \(x\) in the equation and observe the result. Here, \(x^{4} = (-x)^{4} = x^{4}\). As the original equation and the one after substitution are the same, the equation is symmetric about the y-axis. This is the property of an even function.
03

Conclude the result

From the above steps, it is concluded that the function \(y = x^{4}\) is an even function, as it didn't change when we substituted \(-x\) for \(x\) in the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions
Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They are written in the general form:
  • For one variable: \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \)
  • For multiple variables: \( f(x, y) = a_{mn} x^m y^n + a_{m-1,n} x^{m-1} y^n + \ldots + a_{00} \)
Polynomial functions are important because they serve as approximations of other functions and can model a variety of real-world scenarios. A key feature of these functions is their degree, which is determined by the highest power of the variable(s). Higher-degree polynomials can create more complex and interesting graphs. The function given in the exercise, \( y = x^4 \), is a fourth-degree polynomial with one term, known as a monomial.
Function Symmetry
Function symmetry refers to how a graph reflects across an axis or origin. Understanding symmetry helps us quickly identify the nature of the function: whether it is even, odd, or neither. For even functions:
  • An even function is symmetric with respect to the y-axis.
  • This means that for any value \( x \), the function satisfies \( f(x) = f(-x) \).
Odd functions, on the other hand:
  • Demonstrate symmetry about the origin.
  • For these functions, \( f(x) = -f(-x) \).
A function that is neither even nor odd shows no clear symmetry. The function \( y = x^4 \), as shown in the original exercise, is even because substituting \(-x\) yields the same value as the original function, confirming its y-axis symmetry.
Power Functions
Power functions are a type of polynomial function where the variable part is solely raised to a single power and multiplied by a coefficient, typically expressed as \( f(x) = ax^n \). They focus on one "dominant" term due to the power function characteristic, making them simpler than multi-term polynomials. These functions are crucial in understanding growth patterns and scaling behavior:
  • The value of \( n \) significantly influences the shape of the graph.
  • If \( n \) is even, the graph is usually a U-shape; for odd \( n \), the graph tends to have an S-shape.
The given function \( y = x^4 \) is a classic example of a power function, with \( n = 4 \), which results in a symmetrical, upward-opening curve. It helps illustrate how exponents affect the curvature and symmetry of the graph.

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