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. To determine if a function is even, check if \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \).
2. To determine if a function is odd, check if \( f(-x) = -f(x) \) for all \( x \) in the domain of \( f \).
3. For the function \( f(x) = \frac{1}{x} \), calculate \( f(-x) \): \( f(-x) = \frac{1}{-x} = -\frac{1}{x} \).
4. Compare \( f(-x) \) with \( f(x) \) and \( -f(x) \): \( f(-x) = -f(x) \), which satisfies the condition for an odd function.
5. Since \( f(x) = \frac{1}{x} \) is an odd function, its graph exhibits rotational symmetry about the origin.

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 it meets the condition 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.

Graphical Symmetry

Graphical symmetry refers to the way a graph behaves under transformations. An even function exhibits y-axis symmetry, meaning that if you fold the graph along the y-axis, both halves match. An odd function shows rotational symmetry about the origin, indicating that rotating the graph 180 degrees around the origin results in the same graph.

Reciprocal Function

The function ƒ(x) = 1/x is known as a reciprocal function, which is defined for all x except zero. Its graph consists of two branches in the first and third quadrants, approaching the axes but never touching them. Analyzing the behavior of reciprocal functions helps in understanding their symmetry properties and overall shape.