Problem 75 - College Algebra and Calculus: An Applied Approach

Question

State whether the function is even, odd, or neither. $$ g(t)=2 t^{5}-3 t^{2} $$

Step-by-Step Solution

Answer

The function \( g(t) \) is odd.

Step 1: Check if the function is even

An even function satisfies the condition \( g(-t) = g(t) \). Substituting \( -t \) into \( g(t) \), we have: \( g(-t)= 2(-t)^5 - 3(-t)^2 = -2t^5 - 3t^2 \). This is different from the original function \( g(t) = 2t^5 - 3t^2 \), hence, the function is not even.

Step 2: Check if the function is odd

An odd function satisfies the condition \( g(-t) = -g(t) \). Substituting \( -t \) into \( g(t) \), we find that \( g(-t)= -2t^5 - 3t^2 \), which indeed equals \( -g(t) \), hence, the function is odd.

Key Concepts

Understanding Polynomial Functions

Polynomial functions are algebraic expressions made up of terms called monomials. Each term consists of a constant multiplied by a variable raised to a whole number power. For example, in our function \( g(t) = 2t^5 - 3t^2 \), the terms are \( 2t^5 \) and \( -3t^2 \).

Key characteristics of polynomials include the degree, which is the highest power of the variable in the expression. In \( g(t) \), the degree is 5 due to the term \( 2t^5 \).

• The degree of the polynomial can provide information about the potential number of roots or intercepts with the x-axis.
• Polynomials can have varying degrees and must have whole number exponents.

Recognizing a polynomial function is essential as it often simplifies understanding and solving mathematical problems.

Exploring Function Symmetry

Function symmetry is a crucial concept in determining whether a function is even, odd, or neither. The symmetry of a function essentially describes how it behaves when the input values change.

An even function has symmetric behavior about the y-axis, which means \( f(-x) = f(x) \), producing a mirrored effect. For example, a simple even function is \( f(x) = x^2 \) because \( f(-x) = (-x)^2 = x^2 \). This symmetry means the function's graph looks the same on both sides of the y-axis.

• Even functions often have terms with even powers of variables.

On the other hand, odd functions have rotational symmetry about the origin, which means \( f(-x) = -f(x) \). The function \( g(t) = 2t^5 - 3t^2 \) is considered an odd function because when we replace \( t \) with \( -t \), we get \( g(-t) = -2t^5 - 3t^2 \) which equals \(-g(t)\).

• Odd functions typically have terms with odd powers of variables.

Understanding function symmetry helps determine the nature of the function and how it can be represented graphically.

Deconstructing Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operational symbols that together define mathematical calculations. They form the foundation of polynomial functions and various types of equations.

Each part of an algebraic expression can be called a term, and within each term, components like coefficients and variable parts are present. In the expression \( 2t^5 - 3t^2 \):

• The coefficient of \( t^5 \) is 2, and the coefficient of \( t^2 \) is -3.
• The variables \( t \) are raised to a power indicating the degree or order of the term.

Breaking down algebraic expressions into simpler parts is invaluable. This process reveals each part's role in influencing the function's overall behavior and helps in operations such as simplification or determining symmetry.