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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.1.9
Chapter 7, Problem 7.1.9

In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 1 - x/2, x ≤ 0
x/(x + 2), x > 0

Verified step by step guidance
1
Understand that a function is one-to-one if and only if it passes the Horizontal Line Test, meaning no horizontal line intersects the graph more than once.
Analyze the first piece of the function: \(f(x) = 1 - \frac{x}{2}\) for \(x \leq 0\). This is a linear function with a negative slope, which is strictly increasing or decreasing on its domain, so it is one-to-one on \((-\infty, 0]\).
Analyze the second piece of the function: \(f(x) = \frac{x}{x + 2}\) for \(x > 0\). Consider the behavior and monotonicity of this rational function on \((0, \infty)\) to determine if it is one-to-one there.
Check the values of the function at the boundary point \(x = 0\) to ensure continuity or to understand how the two pieces connect, which can affect the overall one-to-one property.
Combine the results from both pieces and verify if any horizontal line intersects the graph in more than one place across the entire domain. If no such horizontal line exists, the function is one-to-one.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

One-to-One Function

A function is one-to-one (injective) if each output corresponds to exactly one input. This means no two different inputs produce the same output. Checking if a function is one-to-one helps determine if it has an inverse function.
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Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain. Understanding how each piece behaves separately and how they connect is essential to analyze the overall function's properties, such as continuity and injectivity.
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Piecewise Functions

Graphical Test for One-to-One Functions

The Horizontal Line Test is used to determine 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. This test is especially useful for piecewise functions.
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One-Sided Limits
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