Problem 85 - Algebra and Trigonometry

Question

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If one branch of a hyperbola is removed from a graph. then the branch that remains must define \(y\) as a function of \(x\)

Step-by-Step Solution

Answer

The statement as given is false. The correct statement should be: If one branch of a hyperbola is removed and the absolute value is taken of the remaining branch, then it will define y as a function of x.

Step 1: Understanding Hyperbolas

A hyperbola is a type of curve that is open-ended and has two distinct branches. It does not pass the vertical line test, meaning it's not a function.

Step 2: Analyzing the Statement

The statement proposes that by removing one branch of the hyperbola, the remaining part would define y as a function of x. This is incorrect as even a single branch of a hyperbola doesn't pass the vertical line test and hence doesn't represent a function.

Step 3: Correcting the Statement

To modify the statement to make it true, the condition must be added that the absolute value is taken of the remaining branch. The absolute value enforces a one-to-one correspondence between x and y. The statement then becomes: If one branch of a hyperbola is removed and the absolute value is taken of the remaining branch, then it will define y as a function of x.

Key Concepts

Function

A function in mathematics is a special kind of relation between two variables, typically represented as \(y = f(x)\). In a function, every input \(x\) is related to exactly one output \(y\). This unique relationship is what defines a true function.
For example, consider the function \(y = 2x + 3\). Here, whatever value you choose for \(x\), there will be a specific, corresponding value of \(y\). Therefore, it meets the criteria of a function because it establishes a one-to-one relationship between \(x\) and \(y\).
It’s important to note that while many mathematical expressions create a relationship between \(x\) and \(y\), not all of them qualify as functions. The notable exceptions include certain curves like hyperbolas, when considered in their entirety.

Vertical Line Test

The vertical line test is a simple visual way to determine if a graph represents a function. The idea is straightforward: take a vertical line and slide it across the graph.

• If at any point, the vertical line only touches the graph at one single point, the relation is a function.
• If the vertical line touches the graph at more than one point at any position, then that graph is not representing a function.

For example, a vertical line on the graph of a hyperbola will usually intersect the graph at two points simultaneously. Hence, a complete hyperbola does not satisfy the vertical line test. This means it is not a function in its entirety.

Curve

When we talk about curves in mathematics, we refer to any line on a graph that isn’t straight. Curves can be open or closed and come in various shapes, such as circles, ellipses, and parabolas. Hyperbolas are also curves, characterized by their two opposing, symmetrical branches.

• In the context of a hyperbola, each branch of the curve is distinct and never intersects with the other.
• The branches are typically symmetrical along the axes, creating the unique open-ended curve of a hyperbola.

These curves are specified by equations like \(x^2/a^2 - y^2/b^2 = 1\). Each equation defines the geometric properties that the curve must adhere to, making it mathematically describable.

Branches

The branches of a hyperbola are the two separate parts of its graph, often mirrored around the center. Each branch stretches to infinity, approaching two imaginary lines called asymptotes but never actually touching them.
When a branch is discussed, it suggests only one of these parts, not the complete hyperbola.

• Even when considering one branch alone, the vertical line test comes into play, showing that one branch doesn't qualify as a function by itself unless transformed, for instance, by taking its absolute value.
• This transformation alters the relationship, potentially allowing it to pass the vertical line test by ensuring that every \(x\) aligns uniquely with a \(y\).

In practice, the concept of branches helps in visualizing and comprehending more complex mathematical relationships present in hyperbolas.