Problem 80 - Algebra and Trigonometry

Question

\(75-82\) . Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$ f(x)=3 x^{3}+2 x^{2}+1 $$

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Define Even and Odd Functions

An even function satisfies the condition \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \). An odd function satisfies \( f(-x) = -f(x) \) for all \( x \) in the domain of \( f \). Our task is to check these conditions for the given function.

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

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

Step 3: Compare \( f(x) \) and \( f(-x) \) to Check Evenness

Compare \( f(x) = 3x^3 + 2x^2 + 1 \) with \( f(-x) = -3x^3 + 2x^2 + 1 \). Since \( f(-x) \) is not equal to \( f(x) \), the function is not even.

Step 4: Compare \( f(x) \) and \( f(-x) \) to Check Oddness

Compare \( f(-x) = -3x^3 + 2x^2 + 1 \) with \( -f(x) = -(3x^3 + 2x^2 + 1) = -3x^3 - 2x^2 - 1 \). Since \( f(-x) \) is not equal to \( -f(x) \), the function is not odd.

Step 5: Conclusion

The function \( f(x) = 3x^3 + 2x^2 + 1 \) is neither even nor odd, as it does not satisfy the conditions for either type of symmetry.

Key Concepts

Function Symmetry

Understanding function symmetry is crucial in mathematics to determine how a function behaves when its input is reversed.
Symmetric functions follow specific rules:

• Even functions: These have symmetry about the y-axis. Mathematically, a function is even if for every x in its domain, the relation \( f(-x) = f(x) \) holds true. Graphically, you could fold the graph along the y-axis and both sides would align perfectly.
• Odd functions: These have symmetry about the origin. An odd function satisfies \( f(-x) = -f(x) \). This means if you rotate the graph 180 degrees around the origin, the graph will look the same.

For the given exercise, we tested if the function \( f(x) = 3x^3 + 2x^2 + 1 \) was even or odd by checking these conditions. Neither condition was met, so we concluded that the function is neither even nor odd.

Polynomial Functions

Polynomial functions form a significant part of algebra and calculus because of their simplicity and versatility. They are expressions constructed from variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
A polynomial function can be expressed in the form:\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]where \( a_n, a_{n-1}, \ldots, a_0 \) are constants, and \( n \) is a non-negative integer.

• Degrees of polynomial: The degree is the highest power of the variable x. For instance, the polynomial \( 3x^3 + 2x^2 + 1 \) has a degree of 3.
• Dominant term: In polynomial functions, the term with the highest degree plays a major role in the overall shape and end behavior of the graph.

Knowing the degree and dominant term can help predict symmetry and other important characteristics of the graph.

Function Evaluation

Function evaluation is a straightforward, yet vital operation in mathematics that involves determining the output of a function for a given input. In simple terms, it's about plugging values into the function equation to get a result.
Consider the function \( f(x) = 3x^3 + 2x^2 + 1 \). To evaluate the function, you substitute the specific value into every occurrence of x.

• If you need \( f(2) \): \[ f(2) = 3(2)^3 + 2(2)^2 + 1 = 24 + 8 + 1 = 33 \] The output is 33.
• Evaluate at \( x = -1 \): \[ f(-1) = 3(-1)^3 + 2(-1)^2 + 1 = -3 + 2 + 1 = 0 \] Here, the function outputs 0.

Function evaluation helps verify function properties and sketch graphs by providing specific points to plot on the coordinate plane.