Provide a short answer to each question. Is ƒ(x)=1/x² an even or an odd function? What symmetry does its graph exhibit?

Verified step by step guidance

1. Recall the definitions: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain, and it is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain.
2. Given \( f(x) = \frac{1}{x^2} \), compute \( f(-x) \) by substituting \( -x \) into the function: \( f(-x) = \frac{1}{(-x)^2} \).
3. Simplify \( f(-x) \): since \( (-x)^2 = x^2 \), we have \( f(-x) = \frac{1}{x^2} \).
4. Compare \( f(-x) \) with \( f(x) \): since \( f(-x) = f(x) \), the function \( f(x) = \frac{1}{x^2} \) is an even function.
5. Because \( f(x) \) is even, its graph exhibits symmetry about the y-axis.

Key Concepts

Even and Odd Functions

A function f(x) is even if f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. It is odd if f(-x) = -f(x), indicating symmetry about the origin. Determining whether a function is even, odd, or neither helps understand its symmetry properties.

Function Symmetry

Symmetry in functions refers to how their graphs mirror across certain lines or points. Even functions exhibit y-axis symmetry, so the left and right sides of the graph are mirror images. Odd functions have origin symmetry, meaning rotating the graph 180 degrees about the origin leaves it unchanged.

Evaluating f(-x) for Rational Functions

To test symmetry for functions like f(x) = 1/x², substitute -x into the function and simplify. For rational functions, this often involves powers of x; even powers yield positive results, affecting symmetry. This substitution helps determine if the function is even, odd, or neither.