Domanda del libro di testo
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
53. lim (x → 1⁺) x^(1/(1 - x))
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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 7.1.33
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
Guida verificata passo dopo passo1
Start by writing the function explicitly: \(f(x) = x^{2} - 2x\) with the domain \(x \leq 1\).
To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = x^{2} - 2x\).
Next, solve the equation \(y = x^{2} - 2x\) for \(x\) in terms of \(y\). Rearrange it as a quadratic equation: \(x^{2} - 2x - y = 0\).
Use the quadratic formula to solve for \(x\): \(x = \frac{2 \pm \sqrt{(-2)^{2} - 4 \cdot 1 \cdot (-y)}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 + 4y}}{2} = 1 \pm \sqrt{1 + y}\).
Since the original domain is \(x \leq 1\), choose the branch of the solution that satisfies this domain (which will be \(x = 1 - \sqrt{1 + y}\)). Then express the inverse function as \(f^{-1}(x) = 1 - \sqrt{1 + x}\). Finally, determine the domain and range of \(f^{-1}\) by considering the range and domain of \(f\), respectively.
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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 within their domains. 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 Restrictions
To ensure a function has an inverse, it must be one-to-one, often requiring domain restrictions. The domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹. Identifying these sets is crucial for correctly defining and verifying the inverse.
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Percorso guidato
5:10Finding the Domain and Range of a Graph
Verification of Inverse Functions
Verifying an inverse involves composing the function and its inverse in both orders: f(f⁻¹(x)) and f⁻¹(f(x)). Both compositions should simplify to x within the appropriate domains. This confirms the correctness of the inverse function and the domain-range assignments.
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4:49Inverse Cosine
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