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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.1.67b
Capitolo 7, Problema 7.1.67b

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:


b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3

Guida verificata passo dopo passo
1
Start with the given function: \(y = \sqrt{3x - 2}\), where \(\frac{2}{3} \leq x \leq 4\).
To find the inverse function, first express the equation in terms of \(x\): square both sides to eliminate the square root, giving \(y^2 = 3x - 2\).
Next, solve for \(x\) by isolating it on one side: add 2 to both sides to get \(y^2 + 2 = 3x\), then divide both sides by 3 to obtain \(x = \frac{y^2 + 2}{3}\).
Define the inverse function \(g\) as \(g(y) = \frac{y^2 + 2}{3}\), which expresses \(x\) as a function of \(y\).
Verify the domain and range of \(g\) to ensure it matches the original function's range and domain, considering the original domain \(\frac{2}{3} \leq x \leq 4\).

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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 g(y) satisfies g(f(x)) = x. Finding the inverse involves solving y = f(x) for x in terms of y, ensuring the function is one-to-one on the given domain.
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Domain and Range Restrictions

To have an inverse, a function must be one-to-one, often requiring domain restrictions. The given interval 2/3 ≤ x ≤ 4 ensures the function is invertible by limiting its domain. Understanding these restrictions is crucial to correctly define and find the inverse function.
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Derivative and Tangent Line Approximation

The derivative of a function at a point gives the slope of the tangent line there, which can approximate the function near that point. For inverse functions, the derivative of the inverse at a point relates to the reciprocal of the original function's derivative, aiding in tangent line calculations.
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