Determine whether each function graphed or defined is one-to-one. y = -√x+5

Verified step by step guidance

1. Understand the definition of a one-to-one function: 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 function given: \( y = -\sqrt{x} + 5 \). This is a transformation of the square root function.
3. Visualize or sketch the graph of \( y = -\sqrt{x} + 5 \). The graph of \( y = \sqrt{x} \) is a curve starting at the origin and increasing to the right. The negative sign reflects it over the x-axis, and the +5 shifts it up by 5 units.
4. Apply the Horizontal Line Test: Draw horizontal lines across the graph. If any horizontal line intersects the graph more than once, the function is not one-to-one.
5. Analyze the graph: Since the graph of \( y = -\sqrt{x} + 5 \) is a downward curve starting from the point (0,5) and moving to the right, each horizontal line will intersect the graph at most once, indicating the function is one-to-one.

Key Concepts

One-to-One Function

A one-to-one function is a type of function where each output value is associated with exactly one input value. This means that no two different inputs produce the same output. To determine if a function is one-to-one, one can use the horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one.

Square Root Function

The square root function, denoted as y = √x, is defined for non-negative values of x and produces non-negative outputs. The graph of the square root function is a curve that starts at the origin and increases gradually. When transformed, such as in the function y = -√x + 5, the graph reflects across the x-axis and shifts vertically, affecting its one-to-one nature.

Graph Interpretation

Graph interpretation involves analyzing the visual representation of a function to understand its properties, such as continuity, intercepts, and whether it is one-to-one. By examining the shape and behavior of the graph, one can determine key characteristics, including whether it passes the horizontal line test, which is essential for identifying one-to-one functions.