Problem 77 - Algebra and Trigonometry

Question

Even and Odd Power Functions What must be true about the integer \(n\) if the function $$f(x)=x^{n}$$ is an even function? If it is an odd function? Why do you think the names "even" and "odd" were chosen for these function properties?

Step-by-Step Solution

Answer

"Even" and "odd" describe the parity of the power \( n \): even powers for even functions and odd powers for odd functions.

Step 1: Understanding Even Functions

A function is called an even function if it satisfies the condition \( f(x) = f(-x) \) for all \( x \) in its domain. For the function \( f(x) = x^n \) to be even, this means \( x^n = (-x)^n \).

Step 2: Establish Condition for Even Function

Since \( (-x)^n = x^n \) when \( n \) is even, the equation holds only for even powers of \( n \). This is because when \( n \) is even, the negative sign cancels out due to the even power, leading to the same expression as \( x^n \).

Step 3: Understanding Odd Functions

A function is called an odd function if it satisfies the condition \( f(-x) = -f(x) \) for all \( x \) in its domain. For the function \( f(x) = x^n \), this means \( (-x)^n = -x^n \).

Step 4: Establish Condition for Odd Function

The equation \( (-x)^n = -x^n \) holds when \( n \) is odd because raising a negative number to an odd power results in a negative number, thereby making \( (-x)^n = -x^n \).

Step 5: Relation to Names "Even" and "Odd"

The names "even" and "odd" were chosen because they describe the parity of the power \( n \) in \( f(x) = x^n \). An even function results from even powers, while an odd function corresponds with odd powers.

Key Concepts

Even Powers

When we talk about "even powers," we are referring to cases where the exponent in a function is an even integer, such as 2, 4, 6, etc. A function expressed as \( f(x) = x^n \) is termed an "even function" if it satisfies the symmetry condition \( f(x) = f(-x) \) for all \( x \) in its domain. This simply means that the graph of the function is symmetric with respect to the y-axis.

• If \( n \) is an even number, then \( (-x)^n = x^n \) because multiplying a negative number an even number of times results in a positive value.
• This symmetry makes even functions mirror images on either side of the y-axis.

Examples of functions with even powers include \( x^2 \) or \( x^4 \). Notice how, regardless of the negative or positive input of \( x \), the output remains the same due to the even exponent, leading to identical values on both sides of the y-axis of the graph.

Odd Powers

"Odd powers" involve exponents that are odd integers, such as 1, 3, 5, etc. A function \( f(x) = x^n \) is named an "odd function" if it meets the condition \( f(-x) = -f(x) \) for every \( x \) in its domain. This means the graph of the function shows origin symmetry, resembling a 180-degree rotation about the origin.

• When \( n \) is odd, \( (-x)^n = -x^n \) holds because a negative base raised to an odd power results in a negative product.
• This rotational symmetry about the origin is a hallmark of odd functions.

Some familiar examples are \( x \) or \( x^3 \). In these cases, the output changes sign when \( x \) does, hence following the pattern of odd functions, where negative \( x \) inputs lead to negative outputs and positive inputs lead to positive outputs.

Function Parity

The concept of "function parity" involves determining whether a function is even, odd, or neither. In mathematical terms, parity relates to how functions behave under transformations. These transformations are either reflections across the y-axis for even functions or rotations about the origin for odd functions.

• Even functions satisfy \( f(x) = f(-x) \), indicative of y-axis symmetry.
• Odd functions satisfy \( f(-x) = -f(x) \), showcasing origin symmetry.
• Parity helps categorize functions and predict their graphical symmetries.

Interestingly, the terminology "even" and "odd" functions stems directly from the parity of the exponents in functions like \( f(x) = x^n \). Functions don't have to be strictly even or odd; many functions fall into neither group, showing different symmetries or no regular patterns at all. Understanding function parity is vital for analyzing the function's graph and its overall behavior.