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Chapter 1: Problem 8

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

Short Answer

Expert verified
The inverse function of \(f(x)=\frac{1}{3}x\) is \(f^{-1}(x) = 3x\), which satisfies the properties \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\).

Step by step solution

01

Find the inverse function

To find the inverse function, replace \(f(x)\) with \(y\), then solve the equation for \(x\).\nSo, \(y=\frac{1}{3}x\) can be rewritten as \(x = 3y\). This implies that the inverse function, \(f^{-1}(x)\), is \(f^{-1}(x) = 3x\).
02

Verify the property \(f\left(f^{-1}(x)\right)=x\)

Substitute \(f^{-1}(x)\) into \(f(x)\). That is, replace \(x\) in \(f(x)\) with \(f^{-1}(x)\), so get \(f\left(f^{-1}(x)\right) = \frac{1}{3}\cdot f^{-1}(x) = \frac{1}{3}\cdot 3x = x\). As \(f\left(f^{-1}(x)\right) = x\), this verifies the first property.
03

Verify the property \(f^{-1}(f(x))=x\)

Substitute \(f(x)\) into \(f^{-1}(x)\). That is, replace \(x\) in \(f^{-1}(x)\) with \(f(x)\), so get \(f^{-1}(f(x)) = 3 \cdot f(x) = 3 \cdot \frac{1}{3}x = x\). As \(f^{-1}(f(x)) = x\), this verifies the second property.

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