Question

Determine algebraically whether each function is even, odd, or neither. \(f(x)=2 x^{4}-x^{2}\)

Short answer

The function is even because \[f(-x) = f(x)\].

Step-by-step solution

Define an Even Function

A function is even if for every x in the domain, the following is true: \[f(-x) = f(x)\]

Define an Odd Function

A function is odd if for every x in the domain, the following is true: \[f(-x) = -f(x)\]

Substitute -x into the Function

Substitute -x into the given function to find \[f(-x)\]: \[f(-x) = 2(-x)^{4} - (-x)^{2}\]

Simplify f(-x)

Simplify \[f(-x)\]:\[f(-x) = 2(x^{4}) - (x^{2}) = 2x^{4} - x^{2}\]

Compare f(-x) and f(x)

Compare the simplified \[f(-x)\] with \[f(x)\]. Notice that \[f(-x) = 2x^{4} - x^{2} = f(x)\]. Therefore, \[f(x)\] is an even function.

Key concepts

function properties

Understanding the properties of a function can help you determine if it is even, odd, or neither. Functions have different characteristics based on how they behave with respect to their inputs and outputs. Some important properties to consider include:

• Symmetry
• Behavior under certain transformations
• Specific values for given inputs

By analyzing these properties, you can categorize the function appropriately. This will be particularly helpful when working with even and odd functions, as their definitions rely heavily on the function's behavior when substituting \(x\) with \(-x\).

even function

An even function exhibits symmetry about the y-axis. This means that for every \(x\) in the domain of the function, the equation \(f(-x) = f(x)\) holds true. In simpler terms, the function's value does not change even if you flip the input's sign.
Here's how you can determine if a function is even:

• Substitute \(-x\) for \(x\) in the original function.
• Simplify the resulting expression.
• Compare the simplified expression to the original function \(f(x)\).

If the expressions match, then the function is even.
For example, in the exercise provided, \(f(x) = 2x^4 - x^2\), substituting \(-x\) gives \(f(-x) = 2(-x)^4 - (-x)^2 = 2x^4 - x^2\). Since \(f(-x) = f(x)\), the function is even.

odd function

An odd function shows rotational symmetry about the origin. In other words, rotating the graph 180 degrees around the origin results in the same graph. For an odd function, \(f(-x) = -f(x)\) holds true for every \(x\) in its domain.
To check if a function is odd, follow these steps:

• Substitute \(-x\) for \(x\) in the original function.
• Simplify the expression.
• Compare the result to \(-f(x)\).

If the simplified expression equals \(-f(x)\), the function is odd.
For example, if a function \(f(x)\) were such that substituting \(-x\) yielded \(-2x^4 + x^2\), and if it equaled \(-f(x)\), we would classify it as an odd function.

substitution in algebra

Substitution is a fundamental concept in algebra that involves replacing a variable with another expression or value. It is extremely useful for testing properties of functions, such as determining if they are even or odd.
For example, consider the function \(f(x) = 2x^4 - x^2\). To determine if it’s even or odd, you substitute \(-x\) into the function:
1. Start with \(f(-x) = 2(-x)^4 - (-x)^2\).
2. Simplify the expression: \(2(x^4) - (x^2) = 2x^4 - x^2\).
This process allows you to see if the substituted function equals the original or its negative counterpart. Mastering substitution will aid in solving complex algebraic problems and enhance your understanding of function behavior.