Determine whether each function graphed or defined is one-to-one. y = -√(100 - x²)

Verified step by step guidance

1. Recall that a function is one-to-one if and only if each output corresponds to exactly one input. This means the function passes the Horizontal Line Test: no horizontal line intersects the graph more than once.
2. Examine the given function: \(y = -\sqrt{100 - x^2}\). Notice that the expression under the square root, \$100 - x^2$, represents a semicircle of radius 10 centered at the origin.
3. Since the square root function outputs only non-negative values, and there is a negative sign in front, \(y\) will be non-positive (zero or negative). This means the graph is the lower semicircle of the circle \(x^2 + y^2 = 100\).
4. Consider the symmetry of the semicircle: for a given \(y\) value (except the lowest point), there are two corresponding \(x\) values (one positive and one negative). This implies the function is not one-to-one because multiple \(x\) values produce the same \(y\) value.
5. Conclude that the function \(y = -\sqrt{100 - x^2}\) is not one-to-one because it fails the Horizontal Line Test due to the semicircular shape of its graph.

Key Concepts

One-to-One Function

A function is one-to-one if each output corresponds to exactly one input, meaning no two different inputs produce the same output. This property ensures the function has an inverse. To test this, the Horizontal Line Test is often used on the graph.

Domain and Range of Functions

The domain is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values). Understanding the domain and range helps determine the behavior of the function and whether it can be one-to-one.

Square Root and Quadratic Functions

The function y = -√(100 - x²) involves a square root of a quadratic expression. The expression inside the root, 100 - x², restricts the domain to values where the radicand is non-negative. The negative sign affects the output values, influencing the function's graph and one-to-one nature.