Problem 98 - College Algebra

Question

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

Step-by-Step Solution

Answer

The solution for \(x\) is \(7\)

Step 1: Simplify the given equation

Start by simplifying the equation \(8+f^{-1}(x-1)=10\), by subtracting 8 from both sides. This gives \(f^{-1}(x-1)=2\)

Step 2: Find the inverse function

To find \(f^{-1}(x-1)\), understand that the inverse of the function \(f(x)\) undoes the action of \(f(x)\). Given \(f(2)=6\), it follows that \(f^{-1}(6)=2\). Therefore, to find \(f^{-1}(x-1)=2\), replace \(2\) by \(6\). This gives \(x-1=6\).

Step 3: Solve for x

Solve the resulting equation \(x-1=6\) for \(x\) by adding 1 to both sides. This gives \(x=7\) as the solution