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Chapter 2: Problem 96

Show that $$f(x)=\frac{3 x-2}{5 x-3}$$ is its own inverse.

Short Answer

Expert verified
The function \(f(x) = \frac{3x-2}{5x-3}\) is its own inverse as demonstrated through substitution and simplification of \(f(f(x))\), which equals \(x\).

Step by step solution

01

Definition of the Function

First, let's list the given function. The function to be proven as its own inverse is \(f(x) = \frac{3x-2}{5x-3}\).
02

Calculate \(f(f(x))\)

To show \(f(x)\) is its own inverse, we need to calculate \(f(f(x))\). That means wherever we see an \(x\), we should replace it with \(f(x)\). So, \(f(f(x))\) equals to \(\frac{3f(x)-2}{5f(x)-3}\). Replace \(f(x)\) with \(\frac{3x-2}{5x-3}\), giving us \(f(f(x)) = \frac{3(\frac{3x-2}{5x-3})-2}{5(\frac{3x-2}{5x-3})-3}\).
03

Simplify \(f(f(x))\)

Next step is to simplify the expression above to get it into the simplest form. After cross multiplication and simplification, we will see that \(f(f(x)) = x\). That proves the function is indeed its own inverse.

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