Problem 60 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=-2 x^{6}-8 x^{2}$$

Step-by-Step Solution

Answer

The given function is even.

Step 1: Understanding Even and Odd Functions

A function is even if, for every x, \(f(-x) = f(x)\). It is odd if, for every x, \(f(-x) = -f(x)\). To determine if the given function \(f(x) = -2x^6 - 8x^2\) is even or odd, we first need to find \(f(-x)\).

Step 2: Substitute -x into the Function

Replace \(x\) with \(-x\) in the function to find \(f(-x)\). So, the expression becomes:\[f(-x) = -2(-x)^6 - 8(-x)^2\]

Step 3: Simplify the Expression

Now, simplify the expression:\((-x)^6 = x^6\) and \((-x)^2 = x^2\). Therefore,\[f(-x) = -2x^6 - 8x^2\]

Step 4: Compare f(-x) with f(x)

We have \(f(-x) = -2x^6 - 8x^2\) and \(f(x) = -2x^6 - 8x^2\). Since \(f(-x) = f(x)\), the function is classified as even.

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics and is essential when working with functions. In simple terms, function evaluation involves substituting a specific value (or set of values) into a function to calculate the corresponding output.
For example, if we have a function \(f(x)\), function evaluation allows us to find \(f(2)\) or \(f(-x)\) by substituting the specific value into the function.
This concept helps us analyze and understand how different inputs affect the function's output.To evaluate functions correctly, follow these steps:

• Identify the expression defining the function.
• Substitute the given value into the expression wherever the variable appears.
• Simplify the expression to find the result.

In the exercise, substituting \(-x\) where \(x\) occurs in the function \(f(x) = -2x^6 - 8x^2\) helped us identify the type of function. Function evaluation is crucial in determining properties such as evenness or oddness.

Even Function

Even functions hold a unique symmetry that makes them particularly interesting. A function is classified as even if it satisfies the condition that for every \(x\), \(f(-x) = f(x)\).
This means that the function is symmetrical around the y-axis and any positive input value will result in the same output as its negative counterpart.Here are some key characteristics of even functions:

• Symmetrical with respect to the y-axis.
• The graph remains unchanged when reflected over the y-axis.
• Common examples include functions like \(y = x^2\) and \(y = \cos(x)\).

In our exercise example, substituting \(-x\) into the function results in \(f(-x) = f(x)\), meaning the function \(f(x) = -2x^6 - 8x^2\) is even.
Understanding even functions is essential as it helps simplify problems and predict behavior without direct evaluation.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers of one or more variables, each multiplied by coefficients.
Polynomial functions are crucial in mathematics as they are simple yet incredibly flexible in modeling and approximating complex systems.The general form of a polynomial function in one variable is:\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \]where:\

• \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants (called coefficients).
• \(n\) is a non-negative integer (degree of the polynomial).
• \(x\) is the variable.

In the given function \(f(x) = -2x^6 - 8x^2\), it is a polynomial with two terms: \(-2x^6\) and \(-8x^2\), making it a 6th-degree polynomial.
Polynomial functions are useful because they are continuous and differentiable, making them suitable for a wide range of applications from physics to engineering.