Question

Determine algebraically whether each function is even, odd, or neither. \(h(x)=3 x^{3}+5\)

Short answer

The function is neither even nor odd.

Step-by-step solution

Define the Function

Start with the given function: \[ h(x) = 3x^3 + 5 \]

Find h(-x)

Compute the function at \( -x \): \[ h(-x) = 3(-x)^3 + 5 \]Since \((-x)^3 = -x^3\), this simplifies to: \[ h(-x) = 3(-x^3) + 5 = -3x^3 + 5 \]

Compare h(x) and h(-x)

Compare \( h(x) = 3x^3 + 5 \) and \( h(-x) = -3x^3 + 5 \).Notice that \( h(x) eq h(-x) \) and \( h(-x) eq -h(x) \).

Draw the Conclusion

Since \( h(x) eq h(-x) \) and \( h(-x) eq -h(x) \), the function is neither even nor odd.

Key concepts

Function Symmetry

Understanding function symmetry is essential for determining if a function is even, odd, or neither. A function is considered even if it is symmetric about the y-axis. This means that for every point \(x, y\), there is a corresponding point \(-x, y\). Algebraically, a function \(f(x)\) is even if \((f(-x) = f(x))\).
On the other hand, a function is odd if it is symmetric about the origin. This means that for every point \(x, y\), there is a corresponding point \(-x, -y\). Algebraically, a function \(f(x)\) is odd if \((f(-x) = -f(x))\).
When a function is neither symmetric about the y-axis nor the origin, it is classified as neither even nor odd. In our exercise, after evaluating \(h(-x)\), we observe neither symmetry, confirming that \((h(x)\)) is neither even nor odd.

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, which involve only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
For example, in the given function \(h(x) = 3x^3 + 5\), we have a third-degree polynomial. Polynomial functions can exhibit various degrees of symmetry. Typically:

• Even-degree polynomials have even symmetry.
• Odd-degree polynomials have odd symmetry.

However, this is not a strict rule for every polynomial. For instance, in our exercise, the polynomial \(3x^3 + 5\) does not follow these rules and is found to be neither even nor odd.

Function Evaluation

Function evaluation involves substituting specific values into the function's formula and simplifying. This is essential to understand how the function behaves when the input changes.
In our problem, we evaluate \(h(x)\) at \(-x\). Start with the original function: \(h(x) = 3x^3 + 5\). By substituting \(-x\), we find:

• Evaluate \((h(-x))\): \((h(-x) = 3(-x)^3 + 5)\).
• Simplify the powers and coefficients: \((h(-x) = -3x^3 + 5)\).

Function evaluation helps us compare \(h(x)\) and \((h(-x)\), allowing a better understanding of symmetry and other properties.

Algebraic Properties

Algebraic properties help in manipulating and simplifying functions. These properties include distributive, associative, and commutative laws among others. When determining symmetry, the algebraic approach involves replacing \(x\) with \(-x\) and simplifying.
In the given exercise, algebraic properties assist us in transforming \((h(-x))\) for comparison:

• Apply the power property: \((-x)^3 = -x^3\).
• Simplify the expression: \((h(-x) = 3(-x^3) + 5 = -3x^3 + 5)\).

Using these properties systematically helps in identifying whether a function is even, odd, or neither effectively.