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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 7.1.21
Chapter 7, Problem 7.1.21

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)=x³-1

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1
Start with the given function: \(y = f(x) = x^3 - 1\).
To find the inverse function \(f^{-1}(x)\), swap the roles of \(x\) and \(y\): write \(x = y^3 - 1\).
Solve this equation for \(y\) to express \(y\) in terms of \(x\): add 1 to both sides to get \(x + 1 = y^3\).
Take the cube root of both sides to isolate \(y\): \(y = \sqrt[3]{x + 1}\).
Therefore, the formula for the inverse function is \(f^{-1}(x) = \sqrt[3]{x + 1}\).

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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. 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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Inverse Cosine

Finding the Inverse Function Algebraically

To find the inverse function f⁻¹(x), start with y = f(x), then solve the equation for x in terms of y. Finally, interchange x and y to express y = f⁻¹(x). This process reverses the roles of dependent and independent variables.
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Inverse Cosine

Graphical Relationship Between f and f⁻¹

The graphs of a function and its inverse are symmetric about the line y = x. Points (a, b) on f correspond to points (b, a) on f⁻¹. This symmetry helps visualize and verify the correctness of the inverse function.
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Determining Concavity from the Graph of f' or f''
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