Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-12: Determine whether \(f\) is even, odd, or neither even nor odd. $$ f(x)=7 x^{5}-4 x^{3} $$

Step-by-Step Solution

Answer

The function \(f(x) = 7x^5 - 4x^3\) is odd.

Step 1: Understand Even and Odd Functions

An even function satisfies the condition \(f(-x) = f(x)\) for all \(x\) in the domain, while an odd function satisfies \(f(-x) = -f(x)\). We need these to determine if \(f(x)=7x^5-4x^3\) is even, odd, or neither.

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

Substitute \(-x\) into the function: \[f(-x) = 7(-x)^5 - 4(-x)^3\] Simplify the expression: \[f(-x) = -7x^5 + 4x^3\].

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

We have \(f(x) = 7x^5 - 4x^3\) and \(f(-x) = -7x^5 + 4x^3\). We observe:- \(f(-x) eq f(x)\) (not even)- \(f(-x) = -f(x)\) since \(-7x^5 + 4x^3 = -(7x^5 - 4x^3)\), (odd)This tells us \(f(x)\) is an odd function.

Key Concepts

Even Functions

Even functions are symmetrically pleasing and have a distinct property that makes them unique. For a function to be classified as even, it must satisfy the rule:

• \(f(-x) = f(x)\) for all \(x\) in the domain.

What does this mean? It suggests that even functions are mirror images with respect to the y-axis. Graphically, if you fold the graph along the y-axis, both sides will match perfectly. A classic example of an even function is the quadratic function \(f(x) = x^2\). When you substitute \(-x\) into the equation, you still get \(x^2\). It's like replacing one mirror image with another, which doesn't change the appearance. This symmetry helps us classify and understand functions better.

Function Properties

Functions can have many defining characteristics, but when we are assessing for evenness or oddness, we focus on the function's symmetry properties. To determine whether a function is even, odd, or neither, you need:

• The formula for the function.
• The ability to substitute \(-x\) for \(x\) and vice versa.

By understanding these properties, it guides us through the evaluation process:
1. Compute \(f(-x)\) by replacing every \(x\) with \(-x\) in the function.
2. Compare \(f(-x)\) to the original function \(f(x)\).

• If \(f(-x) = f(x)\), the function is even.
• If \(f(-x) = -f(x)\), the function is odd.
• If neither condition is met, the function is neither even nor odd.

These properties apply to any function, whether it’s polynomial-based or something altogether different.

Polynomials

Polynomials are fascinating mathematical expressions involving sums of powers of variables. Each term in a polynomial is constructed with a variable raised to a non-negative integer power, multiplied by a coefficient.
An example is \( f(x) = 7x^5 - 4x^3 \), where 7 and -4 are coefficients, and 5 and 3 are powers of \(x\). These powers play a critical role in determining whether the function is even or odd:

• If all powers are even, the polynomial is even.
• If all powers are odd, the polynomial is odd.
• If there is a mix of even and odd powers, the polynomial is often neither.

In the case of \( f(x) = 7x^5 - 4x^3 \), notice both powers, 5 and 3, are odd, leading to the function being classified as odd. Polynomials equip us with a structure to predict function behavior across different mathematical scenarios.

Mathematical Proofs

Mathematical proofs provide the rigorous foundation for understanding and verifying concepts in mathematics. To prove that a function is odd, we follow a structured logic:
1. Start with the definition of an odd function: \(f(-x) = -f(x)\).
2. Substitute \(-x\) into the function to get \(f(-x)\).

• In our example, this resulted in \(f(-x) = -7x^5 + 4x^3\).

3. Compare this with \(f(x)\) to confirm that it satisfies the condition of oddness.

• Here, \(-7x^5 + 4x^3 = -(7x^5 - 4x^3)\), proving \(f(x)\) is an odd function.

Employing proofs ensures that our conclusions are mathematically sound and not based merely on guesswork. They help us solidify our understanding and communicate findings in a robust, reliable way.