Domanda del libro di testo
Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not?
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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 7.1.19
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
Guida verificata passo dopo passo1
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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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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4:49
Inverse Cosine
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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Percorso guidato
5:10Finding the Domain and Range of a Graph
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.
Video consigliato:
4:49Inverse Cosine
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