Determine whether each function graphed or defined is one-to-one. y = -√100 - x^2

Verified step by step guidance

1. Understand that a function is one-to-one if each output value is paired with exactly one input value, meaning it passes the horizontal line test.
2. Consider the given function: \( y = -\sqrt{100 - x^2} \). This is a transformation of the equation of a circle \( x^2 + y^2 = 100 \).
3. Recognize that the function \( y = -\sqrt{100 - x^2} \) represents the lower half of a circle centered at the origin with a radius of 10.
4. Visualize or sketch the graph of the function. The graph is a semicircle below the x-axis, spanning from \( x = -10 \) to \( x = 10 \).
5. Apply the horizontal line test: If any horizontal line intersects the graph more than once, the function is not one-to-one. In this case, any horizontal line below the x-axis will intersect the semicircle at two points, indicating the function is not one-to-one.

Key Concepts

One-to-One Functions

A function is considered one-to-one if it assigns a unique output for every unique input, meaning no two different inputs produce the same output. This can be verified using the horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one.

Graphing Functions

Graphing a function involves plotting its output values against its input values on a coordinate plane. Understanding how to interpret the shape and behavior of the graph is crucial for determining properties like whether the function is one-to-one. The graph of a function can reveal important characteristics such as symmetry and intervals of increase or decrease.

Square Root and Quadratic Functions

The function given, y = -√(100 - x^2), combines a square root and a quadratic expression. The square root function typically produces non-negative outputs, while the negative sign inverts these values. This affects the overall behavior of the function, particularly its range and whether it can be one-to-one, as quadratic functions are generally not one-to-one due to their parabolic shape.