Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not?

Guida verificata passo dopo passo

Step 1: Understand the concept of a one-to-one function. A function is one-to-one if each value of the output (y) corresponds to exactly one value of the input (x). This means the function passes the Horizontal Line Test: no horizontal line intersects the graph more than once.

Step 2: Analyze the given graph of the function \(y = -3x^{3}\). This is a cubic function with a negative leading coefficient, which means it is a decreasing cubic curve passing through the origin.

Step 3: Observe the shape of the graph. Since it is a strictly decreasing curve (it continuously goes down as x increases), it never takes the same y-value twice for different x-values.

Step 4: Apply the Horizontal Line Test. Any horizontal line drawn will intersect the graph at most once, confirming the function is one-to-one.

Step 5: Conclude that the function \(y = -3x^{3}\) is one-to-one because it passes the Horizontal Line Test and each y-value corresponds to exactly one x-value.

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One-to-One Function

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, where no horizontal line intersects the graph more than once. One-to-one functions have inverses that are also functions.

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Horizontal Line Test

The Horizontal Line Test is a graphical method to determine if a function is one-to-one. If any horizontal line crosses the graph more than once, the function is not one-to-one. This test helps verify if the function has a unique inverse.

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Cubic Functions and Their Graphs

Cubic functions like y = -3x³ have an S-shaped curve and are continuous and smooth. The function y = -3x³ is strictly decreasing, meaning it passes the Horizontal Line Test and is one-to-one. Understanding the shape and behavior of cubic functions aids in analyzing their properties.

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