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Chapter 1: Problem 7

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). $$f(x)=6 x$$

Short Answer

Expert verified
The inverse of the function \(f(x)=6x\) is \(f^{-1}(x)=x/6\), which has been verified by the equations \(f(f^{-1}(x))=x\) and \(f^{-1}(f(x))=x\).

Step by step solution

01

Find the Inverse

To find the inverse of the function \(f(x)=6x\), you swap \(x\) and \(y\) and then solve for \(y\). Thus, starting with \(y=6x\) you turn changes it into \(x=6y\). Solving this equation for \(y\) yields \(y=x/6\). Hence, the inverse function \(f^{-1}(x)\) is \(x/6\).
02

Verification 1: \(f(f^{-1}(x))=x\)

Now, verify by plugging \(f^{-1}(x)\) into \(f(x)\). So \(f(f^{-1}(x))=6*(f^{-1}(x))=6*(x/6)=x\). Clearly, \(f(f^{-1}(x))\), simplifies to \(x\) thereby satisfying the equation.
03

Verification 2: \(f^{-1}(f(x))=x\)

To further verify, we plug \(f(x)\) into \(f^{-1}(x)\). So \(f^{-1}(f(x))=f^{-1}(6x)=(6x)/6=x\). Therefore, \(f^{-1}(f(x))\) simplifies to \(x\), so the equation is satisfied.

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