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)=(x+1)², x≥-1

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Start with the given function: \(y = f(x) = (x+1)^2\) with the domain \(x \geq -1\).

To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = (x+1)^2\).

Swap the roles of \(x\) and \(y\) to find the inverse: \(x = (y+1)^2\).

Solve this equation for \(y\): take the square root of both sides to get \(y + 1 = \pm \sqrt{x}\).

Since the original function has domain \(x \geq -1\), the inverse must reflect this restriction, so choose the positive root and solve for \(y\): \(y = \sqrt{x} - 1\). This is the formula for \(f^{-1}(x)\).

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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. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Graphically, the inverse reflects the original function across the line y = x.

Domain and Range Restrictions

To ensure a function has an inverse, it must be one-to-one, often requiring domain restrictions. For f(x) = (x+1)², restricting the domain to x ≥ -1 makes it one-to-one, allowing the inverse to be defined properly and avoiding ambiguity in outputs.

Finding the Inverse Function Algebraically

To find f⁻¹(x), replace f(x) with y, swap x and y, then solve for y. For example, starting with y = (x+1)², swapping gives x = (y+1)², and solving for y yields y = √x - 1, which is the inverse function on the restricted domain.

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)=(x+1)², x≥-1