Question

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

Accepted Answer

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. Examine the given function: $$y = -\sqrt{100 - x^2$}. $$Notice that the expression under the square root, $$100 - x^2$, represents a semicircle of radius 10 centered at the origin. Since the square root function outputs only non-negative values, and there is a negative sign in front, y$ will be non-positive (zero or negative). This means the graph is the lower semicircle of the circle x^2$ + y^2$ = 100. $$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. 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.

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

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Key Concepts

One-to-One Function

Domain and Range of Functions

Square Root and Quadratic Functions