Question

Determine whether the function is even, odd, or neither. $$h(x)=x^5+3 x^4$$

Short answer

The function \(h(x)=x^5+3x^4\) is neither even nor odd.

Step-by-step solution

Substitute -x for x in the function

First, substitute \(-x\) for \(x\) in the function and simplify. This gives: \[ h(-x)=(-x)^5+3(-x)^4 = -x^5+3x^4\]

Compare with original function

Now compare the resulting function with the original function \(h(x)=x^5+3x^4\). The result is different from the original function, so the function is not even.

Check for odd behavior

Switch the sign of \(h(-x)\) and compare with the original function \(h(x)\): \[-h(-x) = -(-x^5+3x^4) = x^5-3x^4\]This is also not the same as the original function, hence the function is not odd either.

Key concepts

Algebraic Functions

Algebraic functions are expressions constructed using a finite number of algebraic operations; these include addition, subtraction, multiplication, division, and taking roots.

For example, the function in the exercise, h(x) = x^5 + 3x^4, is an algebraic function. It is formed by adding two terms, x^5 and 3x^4, which are both monomials and involve the algebraic operations of multiplication and exponentiation.

To analyze functions like these, we may be interested in their properties, such as domain, range, and especially symmetry, which leads to classifying them as even, odd, or neither. An understanding of symmetry is crucial when studying algebraic functions as it can simplify computations and offer insights into their graphical representation.

Function Symmetry

Function symmetry refers to the balanced and proportional arrangement that a function may exhibit on a graph. The two primary types of symmetry relevant to algebraic functions are:

• Even Symmetry: A function f(x) is even if f(-x) = f(x) for all x in the domain. Graphically, this means the function is symmetric about the y-axis.
• Odd Symmetry: A function f(x) is odd if -f(-x) = f(x) for all x in the domain. This implies symmetry about the origin.

In the given exercise, determining the symmetry involved checking these conditions. By substituting -x into the function and comparing it to the original, we established that the function did not meet the criteria for even or odd symmetry, therefore it is classified as neither. The process of testing the function involved straightforward algebraic manipulation, emphasizing the importance of algebraic operations in studying function properties.

Polynomial Functions

Polynomial functions are a type of algebraic function characterized by expressions that are sums of monomial terms, where the exponents are all non-negative integers. The general form of a polynomial function is P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0.

Each term of a polynomial functions consists of a coefficient multiplied by a variable raised to an exponent. These functions are smooth and continuous. The degree of the polynomial, determined by the highest exponent in the expression, influences the function's behavior and the shape of its graph.

The function h(x) = x^5 + 3x^4 from the exercise is a polynomial of degree 5. It showcases the properties of polynomials, such as having a finite number of turning points and end behaviors dependent on the leading term's exponent and coefficient. While analyzing the symmetry, it's interesting to note that polynomial functions with only even powers exhibit even symmetry, and those with only odd powers show odd symmetry—something that doesn't hold for the function in the exercise, as it contains both even and odd powers.