Problem 17 - College Algebra Essentials

Question

Determine whether each function is even, odd, or neither. $$ \begin{aligned} &f(x)=x^{3}+x\\\ &g(x)=x^{2}+x\\\ &h(x)=x^{2}-x^{4}\\\ &f(x)=x^{2}-x^{4}+1\\\ &f(x)=\frac{1}{5} x^{6}-3 x^{2}\\\ &f(x)=x \sqrt{1-x^{2}} \end{aligned} $$

Step-by-Step Solution

Answer

The function \(f(x)=x^3+x\) is odd, \(g(x) = x^2+x\) is neither even nor odd, \(h(x) = x^2 - x^4\) is even, \(f(x)=x^2-x^4+1\) is even, \(f(x) = \frac{1}{5}x^6 - 3x^2\) is even, and \(f(x)=x \sqrt{1-x^2}\) is odd.

Step 1: Analyze \(f(x)=x^{3}+x\)

Substitute \(x\) with \(-x\). So, \(f(-x)=(-x)^{3}+(-x)=-x^{3}-x=-f(x)\). Hence, \(f(x)\) is odd.

Step 2: Analyze \(g(x)=x^{2}+x\)

Substitute \(x\) with \(-x\). So, \(g(-x)=(-x)^{2}+(-x)=x^{2}-x\), which is neither equal to \(g(x)\) nor equal to \(-g(x)\). Hence, \(g(x)\) is neither even nor odd.

Step 3: Analyze \(h(x)=x^{2}-x^{4}\)

Substitute \(x\) with \(-x\). So, \(h(-x)=(-x)^{2}-(-x)^{4}=x^{2}-x^{4}=h(x)\). Hence, \(h(x)\) is even.

Step 4: Analyze \(f(x)=x^{2}-x^{4}+1\)

Substitute \(x\) with \(-x\). So, \(f(-x)=(-x)^{2}-(-x)^{4}+1=x^{2}-x^{4}+1=f(x)\). Hence, \(f(x)\) is even.

Step 5: Analyze \(f(x)=\frac{1}{5}x^{6}-3x^{2}\)

Substitute \(x\) with \(-x\). So, \(f(-x)=\frac{1}{5}(-x)^{6}-3(-x)^{2}=\frac{1}{5}x^{6}-3x^{2}=f(x)\). Hence, \(f(x)\) is even.

Step 6: Analyze \(f(x)=x\sqrt{1-x^{2}}\)

Substitute \(x\) with \(-x\). So, \(f(-x)=(-x)\sqrt{1-(-x)^{2}}=-x\sqrt{1-x^{2}}=-f(x)\). Hence, \(f(x)\) is odd.

Key Concepts

Function Analysis

To determine if a function is even, odd, or neither, we perform function analysis, which involves substituting \(x\) with \(-x\) and observing the behavior of \(f(-x)\). Here's how it works:

• Even Functions: A function is even if \(f(-x) = f(x)\) for all \(x\) in its domain. Graphically, even functions are symmetric with respect to the y-axis.
• Odd Functions: A function is odd if \(f(-x) = -f(x)\) for all \(x\) in its domain. Graphically, these functions have origin symmetry, meaning their graph is symmetric around the origin.
• Neither: If neither condition is satisfied, the function is neither even nor odd.

Checking each function involves directly substituting \(x\) with \(-x\) and then comparing \(f(-x)\) with either \(f(x)\) or \(-f(x)\). This process emphasizes the importance of algebraic manipulation and understanding symmetry in functions.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of \(x\) with coefficients. They form a fundamental part of algebra and calculus.

• General Form: A polynomial function of degree n can be expressed as \({a_n}x^n + {a_{n-1}}x^{n-1} + \dots + {a_1}x + {a_0}\).
• Characteristics: The degree of the polynomial corresponds to the highest power of \(x\). It determines the end behavior of the polynomial as \(x\) approaches infinity or negative infinity.

The determination of whether a polynomial function is even or odd relies on the degrees of the terms:

• Even Polynomials: All terms have even exponents.
• Odd Polynomials: All terms have odd exponents.
• Mixed Polynomials: Contains both even and odd exponents, making them neither even nor odd.

Polynomial functions are important for understanding real-world behavior and performing mathematical modeling.

Mathematical Proofs

Understanding whether a function is even, odd, or neither can be confirmed through mathematical proofs.

• Verification: For each function, calculate \(f(-x)\) and compare it to \(f(x)\) and \(-f(x)\).
• Proof Structure: Begin with the definition, apply algebraic manipulation, and conclude with evidence based on the comparison.

By following these steps, you effectively demonstrate the properties of these functions:

• If \(f(-x) = f(x)\), conclude the function is even and provide an explanation based on y-axis symmetry.
• If \(f(-x) = -f(x)\), conclude the function is odd, supporting it with the concept of origin symmetry.
• If neither holds, conclude the function is neither even nor odd, pointing out the absence of symmetrical properties.

Mathematical proofs provide a rigorous framework to solidify understanding and enhance clarity on these concepts.