Provide a short answer to each question. Is ƒ(x)=1/x 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. Start by finding \( f(-x) \) for the given function \( f(x) = \frac{1}{x} \). Substitute \( -x \) into the function: \( f(-x) = \frac{1}{-x} = -\frac{1}{x} \).
3. Compare \( f(-x) \) with \( f(x) \) and \( -f(x) \): since \( f(-x) = -\frac{1}{x} = -f(x) \), the function satisfies the condition for being an odd function.
4. Because \( f(x) \) is odd, its graph exhibits symmetry about the origin. This means if you rotate the graph 180 degrees around the origin, it looks the same.
5. Summarize: \( f(x) = \frac{1}{x} \) is an odd function and its graph has origin symmetry.

Key Concepts

Even and Odd Functions

A function is even if f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. A function is odd if f(-x) = -f(x), indicating symmetry about the origin. Determining whether a function is even or odd helps understand its symmetry properties.

Function Symmetry

Symmetry in functions refers to how their graphs mirror across certain lines or points. Even functions have y-axis symmetry, while odd functions have origin symmetry. Recognizing symmetry helps in graphing and analyzing function behavior efficiently.

Reciprocal Function Characteristics

The function f(x) = 1/x is a reciprocal function defined for all x ≠ 0. It is known to be an odd function because substituting -x yields f(-x) = -1/x = -f(x). Its graph exhibits origin symmetry, with branches in the first and third quadrants.