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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 7.1.23
Chapter 7, Problem 7.1.23

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
Graph showing function y = (x+1)² for x≥-1 and its inverse y = √x - 1, with labeled axes and curves.

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1
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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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 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.
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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.
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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.
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