Domanda del libro di testo
Evaluate the integrals in Exercises 53–76.
75. ∫y dy/√(1-y^4)
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.1.25
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) = x⁵
Guida verificata passo dopo passo1
Start with the given function: \(f(x) = x^5\). Our goal is to find the inverse function \(f^{-1}(x)\), which means we want to express \(x\) in terms of \(y\) where \(y = f(x)\).
Replace \(f(x)\) with \(y\): \(y = x^5\). To find the inverse, solve this equation for \(x\) in terms of \(y\).
Take the fifth root of both sides to isolate \(x\): \(x = \sqrt[5]{y} = y^{1/5}\). This gives the inverse function: \(f^{-1}(x) = x^{1/5}\).
Determine the domain and range of \(f^{-1}(x)\). Since \(f(x) = x^5\) is defined for all real numbers and is one-to-one, its inverse will also have domain and range as all real numbers.
Verify the inverse by checking the compositions: compute \(f(f^{-1}(x)) = (x^{1/5})^5\) and \(f^{-1}(f(x)) = (x^5)^{1/5}\), and confirm both simplify to \(x\).
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Inverse Functions
An inverse function reverses the effect of the original function, swapping inputs and outputs. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Finding the inverse involves solving y = f(x) for x in terms of y.
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4:49
Inverse Cosine
Domain and Range of Functions and Their Inverses
The domain of a function is the set of all possible inputs, while the range is the set of all possible outputs. For inverse functions, the domain and range swap roles: the domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.
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Percorso guidato
5:10Finding the Domain and Range of a Graph
Properties of Power Functions
Power functions like f(x) = x⁵ are continuous and one-to-one over all real numbers, making them invertible. Since x⁵ is strictly increasing, its inverse is the fifth root function, which also has domain and range as all real numbers.
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Percorso guidato
06:21Properties of Functions
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