Problem 57 - A Graphical Approach to College Algebra

Question

Each function is either even or odd. Use \(f(-x)\) to state which situation applies. $$f(x)=x^{6}-4 x^{4}+5$$

Step-by-Step Solution

Answer

The function is even.

Step 1: Substitute -x into Function

To determine if a function is even or odd, we first substitute \( -x \) for \( x \) in the function \( f(x) = x^6 - 4x^4 + 5 \). This yields \( f(-x) = (-x)^6 - 4(-x)^4 + 5\). Simplifying each term: (-x)^6 = x^6 and (-x)^4 = x^4. Thus, \( f(-x) = x^6 - 4x^4 + 5 \).

Step 2: Compare f(-x) to f(x)

Compare the expression obtained from substituting \( -x \), which is \( f(-x) = x^6 - 4x^4 + 5 \), with the original function \( f(x) = x^6 - 4x^4 + 5 \). Notice that these two expressions are identical.

Step 3: Conclusion About the Function

Since \( f(-x) = f(x) \), the function is even. By definition, a function is even if substituting \( -x \) for \( x \) results in the original function, \( f(x) \). This symmetry indicates that the function is even.

Key Concepts

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number exponents. They are one of the most common types of functions encountered in mathematics. In this context, our function is a polynomial function given by \( f(x) = x^6 - 4x^4 + 5 \). This function has:

• A degree of 6, as the highest exponent of \( x \) is 6.
• Terms like \( x^6 \), \(-4x^4 \), and a constant of 5.
• The form \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients.

Polynomials are powerful in modeling various situations due to their simple composition and smooth, continuous graphing nature. Understanding the structure of polynomial functions helps in analyzing their properties, such as evenness or oddness. Such analysis is useful in predicting the symmetry of these functions on a graph.

Function Symmetry

In the context of functions, symmetry plays an important role in classifying them into even or odd categories.

**Even Functions**
Even functions have the property \( f(-x) = f(x) \). This means the function's graph is symmetrical with respect to the y-axis. Graphically, if you fold the graph along the y-axis, the function will match itself perfectly.

**Odd Functions**
Odd functions satisfy \( f(-x) = -f(x) \), showcasing rotational symmetry about the origin. These functions, if rotated 180 degrees around the origin, will coincide with their original position.

**Symmetric Traits**
Learning about function symmetry assists in understanding how functions behave under transformations, which can be a powerful tool in exploring solutions to complex mathematical problems. It also helps to make visual predictions about function behavior.

Substitution Method

The substitution method is a simple mathematical technique used to determine properties of functions, such as symmetry.

**Using Substitution**
To check for even or odd functions, we replace \( x \) with \( -x \) in the function. Our goal is to see if the output equals the original or negation of the original function. For the function \( f(x) = x^6 - 4x^4 + 5 \), substituting \( -x \) gives:

• Substitution yields \( f(-x) = (-x)^6 - 4(-x)^4 + 5 \).
• After simplifying, it turns into \( f(-x) = x^6 - 4x^4 + 5 \).

This result implies that since \( f(-x) = f(x) \), the function is even.

**Effectiveness of Substitution**
Substitution helps in effectively simplifying expressions and verifying functional properties. This method is a staple in many areas of algebra and calculus, providing a foundation for more complex problem-solving strategies.