Problem 62 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=-x^{5}+2 x^{3}-3 x$$

Step-by-Step Solution

Answer

The function \(f(x) = -x^5 + 2x^3 - 3x\) is odd because \(f(-x) = -f(x)\).

Step 1: Evaluate \( f(-x) \)

To determine if the function is even, odd, or neither, substitute \(-x\) in place of \(x\) in the function \(f(x) = -x^5 + 2x^3 - 3x\). This gives us: \[ f(-x) = -(-x)^5 + 2(-x)^3 - 3(-x) \] Calculating each term separately, we find:- \((-x)^5 = -x^5\) so \(-(-x)^5 = x^5\)- \(2(-x)^3 = -2x^3\)- \(-3(-x) = 3x\)

Step 2: Simplify \( f(-x) \)

Now, simplify the expression obtained:\[ f(-x) = x^5 - 2x^3 + 3x \]

Step 3: Compare \( f(-x) \) and \( -f(x) \)

To check for an odd function, compare \( f(x) \) and \(-f(x)\). We compute:\[ -f(x) = -(-x^5 + 2x^3 - 3x) = x^5 - 2x^3 + 3x \] Notice that: \(f(-x) = x^5 - 2x^3 + 3x = -f(x)\) which indicates that \(f(x)\) is an odd function.

Key Concepts

Odd Functions

Odd functions have a unique property. When you evaluate the function at \(-x\), the result is the negative of what you get when you evaluate \(x\). Formally put, a function \(f(x)\) is considered odd if \(f(-x) = -f(x)\).
This special behavior indicates a form of symmetry in the function.

• This symmetry is rotational around the origin: if you rotate the function's graph by 180 degrees about the origin, it looks the same.
• Odd functions reflect across both axes, not just one.

To determine if our function \(f(x) = -x^5 + 2x^3 - 3x\) is odd:

1. We evaluate \(f(-x)\), simplifying the terms to get \(x^5 - 2x^3 + 3x\).
2. We also compute \(-f(x)\), which results in \(x^5 - 2x^3 + 3x\).

Since these two expressions are identical, we confirm that \(f(x)\) is indeed an odd function. Understanding this property helps in graph analyses and solving complex mathematical problems.

Even Functions

Even functions exhibit another form of symmetry. For these functions, when you input the negative of a number, you get the same result as when you input the positive of that number. Mathematically, a function \(f(x)\) is even if \(f(-x) = f(x)\).
This property encases symmetry about the y-axis: reflecting all points across the y-axis results in the same graph.

• This means an even function looks identical on both sides of the y-axis.
• Common examples of even functions include \(x^2\), \(cos(x)\), and \(x^4\).

In our step-by-step solution, we explored the possibility of \(f(x) = -x^5 + 2x^3 - 3x\) being even. Actions such as evaluating \(f(-x)\) and comparing it with \(-f(x)\) lead to the conclusion about the nature of this function.
In exercises, verifying both even and odd symmetry checks ensures complete understanding of function behavior.

Function Evaluation

Function evaluation is a fundamental process in mathematics where given a function \(f(x)\), you replace \(x\) with another value to calculate the function output.
This simple substitution is crucial for identifying function traits, verifying symmetries, and solving equations.

• Input values can be numbers, algebraic expressions, or even other functions.
• Changing inputs helps explore different aspects and apply functions in various problem-solving scenarios.
• In the given problem, evaluating \(f(-x) \) helped us test if the function is even, odd, or neither.

Evaluation involves:

1. Substitute \(-x\) or any value directly into the original expression; remember to correctly handle sign changes and powers.
2. Simplify the expression after substitution to get an equivalent form.

Being meticulous in substitutions allows us not only to classify function types but also enhances graphing skills and calculus readiness.