Question

Decide whether the function is even, odd, or neither. \(f(x)=x^{6}-2 x^{2}+3\)

Short answer

The function \(f(x)=x^{6}-2 x^{2}+3\) is even.

Step-by-step solution

Analyze the given function

The function given is \(f(x)=x^{6}-2 x^{2}+3\). This function is a polynomial function, having an even degree and real coefficients.

Check if the function is even

Substitute \(x\) with \(-x\) in the given equation and simplify. \[f(-x)=(-x)^{6}-2 (-x)^{2}+3 = x^{6}-2 x^{2}+3\] The simplified function is equal to the original function \(f(x)\), so it satisfies the condition of an even function.

Check if the function is odd

If the function was even we don't need to check if it is odd, because a function cannot be both even and odd at the same time. But, for the sake of completeness, let's show that it cannot be odd.\nTo do so, substitute \(x\) with \(-x\) in the given equation and multiply with \(-1\). \[-f(-x) = -[(-x)^{6}-2 (-x)^{2}+3] = -x^{6} + 2x^{2} - 3\] This is not equal to original function \(f(x)\), so it does not satisfy the condition of an odd function.

Key concepts

Polynomial Functions

Polynomial functions are expressions that consist of variables and coefficients, using only addition, subtraction, multiplication, and non-negative integer exponents of variables. A typical polynomial function is of the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n, a_{n-1}, \ldots, a_0\) are constants known as coefficients and \(n\) is a non-negative integer known as the degree of the polynomial.

• The degree of a polynomial function is the highest power of the variable within the expression.
• Polynomial functions can have various shapes on a graph, including lines, parabolas, and more complex curves depending on the degree.
• Higher degree polynomials can have many bends, increasing or decreasing in various sections, which can make their graphs appear quite complex.

Polynomial functions are extensively used in algebra for modeling real-world systems and solving many practical problems. They also provide a foundation for higher-level mathematics topics.

Odd Functions

Odd functions are special types of functions with a unique symmetry property. Specifically, a function \(f(x)\) is considered odd if it satisfies the condition \(f(-x) = -f(x)\) for all values of \(x\). This means that the graph of an odd function is symmetric with respect to the origin.

• Visualize an odd function as a graph that looks the same after being rotated 180 degrees about the origin.
• Examples of odd functions include \(f(x) = x\), \(f(x) = x^3\), and \(f(x) = \sin x\).
• If a polynomial function contains only odd powers of \(x\), the function is odd.

Odd functions hold significance in mathematics due to their symmetry, which can simplify problems in calculus, such as integrating odd functions over symmetric intervals, where the result is zero under certain conditions.

Real Coefficients

Real coefficients refer to the constants in a polynomial function that are real numbers. A polynomial with real coefficients has the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where each \(a_i\) is a real number. Real numbers include all the numbers on the number line, encompassing both rational numbers (like fractions) and irrational numbers (like the square root of 2).

• Real coefficients ensure that the polynomial function is easy to interpret and work with in real-world applications.
• When graphed, polynomial functions with real coefficients will cross the x-axis at real roots or touch it without crossing at repeated roots.
• Polynomials with real coefficients are contrasted with polynomials that might have complex coefficients, which involve imaginary numbers.

Understanding real coefficients is crucial for discussing and solving polynomial equations since it impacts the nature of solutions and their graphical representation.