Determine whether each function graphed or defined is one-to-one. y = 2x - 8

검증된 단계별 안내

Recall that a function is one-to-one if each output value corresponds to exactly one input value. This means the function passes the Horizontal Line Test: no horizontal line intersects the graph more than once.

Given the function \(y = 2x - 8\), recognize that this is a linear function with slope 2 and y-intercept -8.

Since the slope is not zero, the function is strictly increasing, meaning as \(x\) increases, \(y\) increases without repeating any output values.

Because the function is strictly increasing, it passes the Horizontal Line Test, indicating it is one-to-one.

Therefore, the function \(y = 2x - 8\) is one-to-one because it has a unique output for every input.

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주요 개념

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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 that is also a function.

Linear Functions

Linear functions have the form y = mx + b, where m is the slope and b is the y-intercept. They produce straight lines, and their behavior depends on the slope, which affects whether the function is one-to-one.

Horizontal Line Test

The horizontal line test determines if a function is one-to-one by checking if any horizontal line intersects the graph more than once. If it does, the function is not one-to-one; if it doesn't, the function is one-to-one.