Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = xe^x-e^x
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.1.5
Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not?
Verified step by step guidance1
Step 1: Understand the definition of a 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.
Step 2: Analyze the graph of the function \(y = \frac{1}{x}\). Notice that the graph has two branches, one in the first quadrant and one in the third quadrant, and it never touches the x-axis or y-axis (asymptotes).
Step 3: Apply the Horizontal Line Test to the graph. For any horizontal line drawn, it will intersect the graph at most once because the function decreases continuously in the first quadrant and increases continuously in the third quadrant without repeating y-values.
Step 4: Conclude that since the function \(y = \frac{1}{x}\) passes the Horizontal Line Test, it is a one-to-one function.
Step 5: Summarize that the function \(y = \frac{1}{x}\) is one-to-one because each y-value corresponds to exactly one x-value, and the graph confirms this visually.
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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 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. Graphically, a one-to-one function passes the Horizontal Line Test, where no horizontal line intersects the graph more than once.
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Horizontal Line Test
The Horizontal Line Test is a visual 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 quickly identify whether a function has an inverse function.
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Behavior of the Function y = 1/x
The function y = 1/x is defined for all x except zero and has two branches in the first and third quadrants. It is strictly decreasing on each interval (-∞, 0) and (0, ∞), and it passes the Horizontal Line Test on its domain, making it a one-to-one function.
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Related Practice
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Textbook Question
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Textbook Question
135. Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y = e^(2x), below by the curve y = e^x, and on the right by the line x = ln(3).