Provide a short answer to each question. Is ƒ(x)=1/x^2 an even or an odd function? What symmetry does its graph exhibit?

Verified step by step guidance

1. Step 1: Recall the definitions of even and odd functions. A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain. A function is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain.
2. Step 2: Consider the function \( f(x) = \frac{1}{x^2} \). To determine if it is even, substitute \( -x \) into the function: \( f(-x) = \frac{1}{(-x)^2} \).
3. Step 3: Simplify \( f(-x) = \frac{1}{(-x)^2} \) to \( \frac{1}{x^2} \), which is equal to \( f(x) \).
4. Step 4: Since \( f(-x) = f(x) \), the function \( f(x) = \frac{1}{x^2} \) is an even function.
5. Step 5: Even functions exhibit symmetry about the y-axis. Therefore, the graph of \( f(x) = \frac{1}{x^2} \) is symmetric with respect to the y-axis.

Key Concepts

Even and Odd Functions

A function is classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain, indicating symmetry about the y-axis. Conversely, a function is odd if f(-x) = -f(x), which shows symmetry about the origin. Understanding these definitions is crucial for determining the nature of the function ƒ(x) = 1/x^2.

Graphical Symmetry

Graphical symmetry refers to the way a function's graph behaves in relation to specific axes. An even function exhibits y-axis symmetry, meaning that if you fold the graph along the y-axis, both halves match. This concept helps visualize the properties of the function and aids in understanding its behavior.

Reciprocal Functions

Reciprocal functions, such as ƒ(x) = 1/x^2, are defined as the inverse of a variable raised to a power. These functions have unique characteristics, including vertical asymptotes and specific symmetry properties. Recognizing the behavior of reciprocal functions is essential for analyzing their graphs and understanding their symmetry.