Textbook Question
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x² − 2x, x ≤ 1
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 7.1.19
Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.
f(x)=x²+1, x≥0
Verified step by step guidance1
Start with the given function: \(y = f(x) = x^{2} + 1\) where \(x \geq 0\).
To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = x^{2} + 1\).
Swap the variables \(x\) and \(y\) to find the inverse: \(x = y^{2} + 1\).
Solve this equation for \(y\): subtract 1 from both sides to get \(x - 1 = y^{2}\).
Since \(x \geq 0\) in the original function, take the positive square root to get \(y = \sqrt{x - 1}\), which is the formula for \(f^{-1}(x)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Functions
An inverse function reverses the effect of the original function, swapping inputs and outputs. If y = f(x), then x = f⁻¹(y). The graph of an inverse function is a reflection of the original function's graph across the line y = x.
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Domain and Range Restrictions
To find an inverse function, the original function must be one-to-one, often requiring domain restrictions. Here, f(x) = x² + 1 is restricted to x ≥ 0 to ensure it is invertible, as the parabola is not one-to-one over all real numbers.
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Finding the Inverse Function Algebraically
To find f⁻¹(x), replace f(x) with y, swap x and y, then solve for y. For f(x) = x² + 1 with x ≥ 0, swapping gives x = y² + 1, so y = √(x - 1), which matches the inverse function shown in the graph.
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Related Practice
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