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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.1.72b
Chapter 7, Problem 7.1.72b

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.


72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

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1
Start with the given function: \(y = 2 - x - x^{3}\), where \(-2 \leq x \leq 2\).
To find the inverse function \(g\), solve the equation \(y = 2 - x - x^{3}\) for \(x\) in terms of \(y\).
Rewrite the equation as \(x^{3} + x = 2 - y\) to isolate terms involving \(x\) on one side.
Recognize that this is a cubic equation in \(x\): \(x^{3} + x - (2 - y) = 0\).
Use an appropriate method (such as the cubic formula or a computer algebra system) to solve for \(x\) as a function of \(y\), and define this solution as the inverse function \(g(y)\).

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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. 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, which may require algebraic manipulation and checking domain restrictions to ensure the inverse is well-defined.
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Derivative of Inverse Functions

The derivative of an inverse function at a point relates to the derivative of the original function by the formula g'(y) = 1 / f'(x), where y = f(x). This relationship helps find the slope of the tangent line to the inverse function without explicitly differentiating it, provided f'(x) ≠ 0 at the point of interest.
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Derivatives of Other Inverse Trigonometric Functions

Tangent Line Approximation

Tangent line approximation uses the derivative at a point to approximate the function near that point with a linear function. For a function f at x₀, the tangent line is y = f(x₀) + f'(x₀)(x - x₀). This concept extends to inverse functions, allowing approximation of g(y) near y₀ using the slope of the inverse function.
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Related Practice
Textbook Question

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.

b. f(x) = x, g(x) = x², (a, b) arbitrary

Textbook Question

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b. Find all inflection points for f.

Textbook Question

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2. b. e^(kt) = 10

Textbook Question

Find the inverse of f(x)=x+b (b constant). How is the graph of f^(-1) related to the graph of f?

Textbook Question

131. Let f(x) = x * e^(−x).

b. Find all inflection points for f.

Textbook Question

71. Locate and identify the absolute extreme values of cos(ln x) on [1/2, 2]