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Chapter 2: Problem 71

Describe how to find the inverse of a one-to-one function.

Short Answer

Expert verified
To find the inverse of a one-to-one function, reverse the roles of \(x\) and \(y\), solve the new equation for \(y\), and then validate your result. The resulting equation is your inverse function.

Step by step solution

01

Understand the concept of one-to-one function

A function \(f(x)\) is said to be one-to-one if no two different input values will produce the same output. In other words, if \(f(a) = f(b)\), then \(a = b\). This property allows to establish a unique inverse for the function.
02

Swap the roles of \(x\) and \(y\)

Assume that we have a function \(y = f(x)\). The first step in finding its inverse is to switch the roles of \(y\) and \(x\). We rewrite \(y = f(x)\) as \(x = f^{-1}(y)\)
03

Solve the new equation for \(y\)

Now, solve the new equation \(x = f^{-1}(y)\) for \(y\). The solved equation will represent the inverse function \(f^{-1}(x)\). This might involve basic algebraic operations of addition, subtraction, multiplication, division, or solving for a square root as the case may be.
04

Validate your result

Now, let's check the result. A function and its inverse undo each other. Plugging \(f^{-1}(x)\) as the input to the function \(f(x)\), or plugging \(f(x)\) as the input to \(f^{-1}(x)\) should yield \(x\). If it does, then the result is valid.

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Most popular questions from this chapter

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+6 x+2 y+6=0$$

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