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Chapter 1: Problem 9

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). $$f(x)=3 x+1$$

Short Answer

Expert verified
The inverse function of \(f(x)=3x+1\) is \(f^{-1}(x)=(x-1)/3\). All the verifications also confirm it as the inverse function.

Step by step solution

01

Identify the original function

The original function given is \(f(x)=3x+1\). This represents a straight line with a slope of 3 and \(y\)-intercept of 1.
02

Find the inverse function

To find the inverse, we first replace \(f(x)\) with \(y\), so \(y=3x+1\). Then we switch the roles of \(y\) and \(x\) to get \(x=3y+1\). Now we solve for \(y\) to get the inverse function. Subtract 1 from both sides to get \(x-1=3y\), then divide both sides by 3. The inverse function is \(f^{-1}(x)=(x-1)/3\).
03

Confirm the properties of the inverse function

We can confirm that this function is indeed the inverse by checking the compositions \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\). Substituting \(f^{-1}(x)=(x-1)/3\) to \(f(x)\), we get \(f(f^{-1}(x))=3((x-1)/3)+1=x\). Now substitute \(f(x)=3x+1\) to \(f^{-1}(x)\), we get \(f^{-1}(f(x))=(3x+1-1)/3=x\). Both these results equal to \(x\), confirming the inverse.

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