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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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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주요 개념

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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.

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.

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.

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