Problem 7 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-12: Determine whether \(f\) is even, odd, or neither even nor odd. $$ f(x)=8 x^{3}-3 x^{2} $$

Step-by-Step Solution

Answer

The function \(f(x)=8x^3-3x^2\) is neither even nor odd.

Step 1: Recall Definitions

A function is **even** if for every \(x\), \(f(-x) = f(x)\). A function is **odd** if for every \(x\), \(f(-x) = -f(x)\). If neither condition is met, the function is neither even nor odd.

Step 2: Write Expression for \(f(-x)\)

The given function is \(f(x) = 8x^3 - 3x^2\). To find \(f(-x)\), substitute \(-x\) for \(x\): \[ f(-x) = 8(-x)^3 - 3(-x)^2 \] Simplify to get: \[ f(-x) = -8x^3 - 3x^2 \]

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

The expression for \(f(x)\) is \(8x^3 - 3x^2\), and \(f(-x) = -8x^3 - 3x^2\). - Check if \(f(-x) = f(x)\): \( -8x^3 - 3x^2 eq 8x^3 - 3x^2\). - Check if \(f(-x) = -f(x)\): \(-8x^3 - 3x^2 = -(8x^3 - 3x^2) = -8x^3 + 3x^2\). Neither condition is satisfied.

Step 4: Conclusion

Since neither condition for even nor odd function is satisfied, \(f(x) = 8x^3 - 3x^2\) is neither even nor odd.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving variables and coefficients, consisting of one or more terms. These terms are combined by addition, subtraction, and multiplication but never division by a variable. A polynomial function can be written in the general form: \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \]where the coefficients \(a_n, a_{n-1},..., a_1, a_0\) are constants, and \(n\) is a non-negative integer representing the highest power of \(x\).
In the given function \(f(x) = 8x^3 - 3x^2\), the polynomial has two terms:

• \(8x^3\) - a cubic term with a coefficient of 8.
• \(-3x^2\) - a quadratic term with a coefficient of -3.

The degree of the polynomial is determined by the highest exponent, which in this case is 3. Therefore, it is a cubic polynomial, and the overall behavior of \(f(x)\) will be influenced by its highest-degree term.

Function Symmetry

Function symmetry is a key concept when determining whether a function is even, odd, or neither. Symmetrical functions exhibit repeating patterns about the y-axis or the origin, while non-symmetrical functions do not.
For even functions, the definition is that \(f(-x) = f(x)\). This means the function is mirror-symmetrical across the y-axis. Examples of even functions include \(x^2\) and \(\cos(x)\).
For odd functions, the definition is that \(f(-x) = -f(x)\). This means the function is symmetrical about the origin. An example of an odd function is \(x^3\) or \(\sin(x)\).
If a function does not meet either of these criteria, it is neither even nor odd. In the example exercise, after substituting \(-x\) into \(f(x) = 8x^3 - 3x^2\), you get \(-8x^3 - 3x^2\), which does not meet the conditions for symmetry, indicating the function is neither even nor odd. This helps in predicting the graph's layout.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. They are the building blocks for discussing functions and equations in mathematics.
They can range from simple, like \(3x\), to complex, involving multiple terms like in the polynomial \(8x^3 - 3x^2\). Understanding algebraic expressions involves knowing how to manipulate and simplify these terms.
To evaluate \(f(-x)\) in algebraic terms, you substitute \(-x\) for every \(x\) in the expression, as seen in the provided solution:

• Substitute: \(8(-x)^3 - 3(-x)^2\)
• Simplify: \(-8x^3 - 3x^2\)

This process shows how different rules of algebraic manipulation are used to explore the properties of functions, such as symmetry. Simplifying expressions correctly allows you to assess how functions behave and can inform further mathematical analyses.