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Chapter 2: Problem 98

If \(f(2)=6,\) and \(f\) is one-to-one, find \(x\) satisfying \(8+f^{-1}(x-1)=10\).

Short Answer

Expert verified
The solution to the equation is \(x=7\).

Step by step solution

01

Analyzing the given equation

Starting with the given equation \(8+f^{-1}(x-1)=10\), the goal is to isolate \(f^{-1}(x-1)\). Do so by subtracting 8 from both sides of the equality which yields \(f^{-1}(x-1)=2\).
02

Working with the inverse function

Since \(f\) is one-to-one, then \(f(2)=6\), and inversely \(f^{-1}(6)=2\). Also, it is known that \(f^{-1}(x-1)=2\) from step 1. This can imply that \(x-1=6\).
03

Finding the value of \(x\)

In order to find the value of \(x\) from the equation \(x-1=6\), add 1 to both sides of the equation, thus \(x=7\).

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