Branch — Definition, Formula & Examples

A branch is a continuous piece of a curve or graph that is separated from other pieces by a discontinuity, asymptote, or undefined point. For example, a hyperbola has two branches, and the function

y = 1/x

has a branch on each side of the

y

-axis.

A branch of a curve $C$ is a maximal connected subset of $C$ in the plane. For a function $f$ , a branch is a restriction of $f$ to a maximal interval on which $f$ is continuous. In the context of multi-valued relations, each single-valued continuous selection is called a branch.

How It Works

To identify the branches of a curve, look for points where the curve breaks — typically at vertical asymptotes, points of discontinuity, or places where the relation is undefined. Each unbroken, continuous segment is one branch. For a hyperbola

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

, the left half (where

x \leq -a

) and the right half (where

x \geq a

) are the two branches, separated by the gap where no real points exist. When working with inverse trig or square root functions, you often choose a single branch to make the relation a proper function.

Worked Example

Problem: Identify the branches of the hyperbola

\frac{x^2}{9} - \frac{y^2}{4} = 1

Step 1: Solve for

x

to find where points exist on the curve.

x^2 = 9\left(1 + \frac{y^2}{4}\right) \geq 9 \implies |x| \geq 3

Step 2: Since

|x| \geq 3

, no points on the curve have

-3 < x < 3

. This gap separates the curve into two connected pieces.

Step 3: The right branch consists of all points with

x \geq 3

, and the left branch consists of all points with

x \leq -3

Answer: The hyperbola has two branches: the right branch (opening right, vertex at

(3, 0)

) and the left branch (opening left, vertex at

(-3, 0)

Why It Matters

Understanding branches is essential when graphing rational functions and conic sections in precalculus. In calculus, recognizing separate branches helps you correctly evaluate limits and avoid applying continuity theorems across asymptotes.

Common Mistakes

Mistake: Treating both branches of a hyperbola as one continuous curve and drawing a line connecting them through the center.

Correction: The two branches of a hyperbola never meet. There is always a gap between them — no real points exist in that region.