Problem 66 - Algebra and Trigonometry

Question

61–68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = 3x^{3} + {2x}^{2} + {1}$$

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Review Definitions

An even function satisfies \( f(-x) = f(x) \), and an odd function satisfies \( f(-x) = -f(x) \). We will test \( f(x) = 3x^3 + 2x^2 + 1 \) to see if it is even, odd, or neither.

Step 2: Calculate f(-x)

Substitute \(-x\) into the function: \[ f(-x) = 3(-x)^3 + 2(-x)^2 + 1 = -3x^3 + 2x^2 + 1 \].

Step 3: Check for Even Function

Compare \( f(-x) \) with \( f(x) \). For \( f\) to be even, it must satisfy \( f(-x) = f(x) \): \(-3x^3 + 2x^2 + 1 eq 3x^3 + 2x^2 + 1 \). The function is not even.

Step 4: Check for Odd Function

Compare \( f(-x) \) with \(-f(x)\). For \( f\) to be odd, it must satisfy \( f(-x) = -f(x) \): \(-3x^3 + 2x^2 + 1 eq -(3x^3 + 2x^2 + 1) = -3x^3 - 2x^2 - 1 \). The function is not odd.

Step 5: Conclusion on Evenness or Oddness

Since \( f(x) \) is neither equal to \( f(-x) \) nor \(-f(x)\), the function is neither even nor odd.

Key Concepts

Function Symmetry

Symmetry in functions is a fascinating concept that helps us understand how a graph behaves and is shaped. A function can be classified as even, odd, or neither, based on its symmetry.

• An **even function** shows symmetry about the y-axis. This means that for every point (x, y) on the graph, there is a corresponding point (-x, y). The algebraic test for even functions is given by the equation \( f(-x) = f(x) \).
• On the other hand, an **odd function** presents symmetry around the origin. Simply stated, if you were to rotate the graph 180 degrees around the origin, it would look the same. The mathematical test for an odd function is \( f(-x) = -f(x) \).

For our specific function \( f(x) = 3x^3 + 2x^2 + 1 \), after evaluating the function with \(-x\), we found that it does not satisfy the criteria for either even or odd symmetry. Understanding these definitions is crucial for anyone looking to master graphing and function analysis.

Polynomial Functions

Polynomial functions are expressions composed of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents. They are fundamental in mathematics and allow us to understand complex relationships through familiar terms.
Polynomials are classified by their degree, which is the highest power of the variable in the expression. For instance:

• The function \( f(x) = 3x^3 + 2x^2 + 1 \) is a cubic polynomial, as the highest exponent of \( x \) is 3.
• Polynomial functions can have terms that are even, odd, or neither based on their exponents. Here, the term \( 3x^3 \) is odd since 3 is odd, and \( 2x^2 \) is even since 2 is even.

Because polynomial functions can include a blend of even and odd terms, the overall function may not necessarily fulfill the criteria for evenness or oddness as seen in \( f(x) = 3x^3 + 2x^2 + 1 \). This composite nature makes analyzing polynomial functions both intriguing and insightful.

Graph Sketching

Graph sketching is a valuable skill that enables you to visualize mathematical functions and their properties. Being able to sketch a graph provides a deeper understanding of the function's behavior and symmetry.
For the function \( f(x) = 3x^3 + 2x^2 + 1 \), sketching can be approached by considering its individual terms:

• The **term \( 3x^3 \)** shapes the graph with a cubic curve that extends from bottom-left to top-right, indicative of odd functions.
• The **term \( 2x^2 \)** adds a parabolic upward U-shape to the graph, characteristic of even functions.
• Meanwhile, **the constant 1 term** shifts the entire graph upwards by one unit.

Though \( f(x) = 3x^3 + 2x^2 + 1 \) is neither purely even nor odd, the combination of these terms results in a lively graphical representation. Graph sketching not only aids in comprehending how functions interact within an equation but also prepares you to solve real-world problems by visualizing mathematical concepts effectively.