Problem 66 - College Algebra

Question

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

Step-by-Step Solution

Answer

The function \(f(x)=\frac{3x-2}{5x-3}\) is its own inverse.

Step 1: Write down the function and its inverse

Let's start by writing the original function: \(f(x)=\frac{3x-2}{5x-3}\). Assuming that this function is its own inverse, we can write the inverse as: \(f^{-1}(x)=\frac{3x-2}{5x-3}\)

Step 2: Check if the inverse undone operations of the original function

We can check if a function is its own inverse by substituting the inverse function into the original function. For \(f(f^{-1}(x))\), if we place \(f^{-1}(x)\) into \(f(x)\) we get: \(f(f^{-1}(x)) = f\left(\frac{3x-2}{5x-3}\right)\). Simplifying this should result in \(x\) if the function is its own inverse.

Step 3: Calculation

Substitute into the original function: \(f\left(\frac{3x-2}{5x-3}\right) = \frac{3(\frac{3x-2}{5x-3})-2}{5(\frac{3x-2}{5x-3})-3}\). Simplify the numerator and denominator separately: \(\frac{3\cdot\frac{3x-2}{5x-3}-2}{5 \cdot \frac{3x-2}{5x-3}-3} = \frac{\frac{9x-6}{5x-3}-2}{\frac{15x-10}{5x-3}-3} = \frac{\frac{9x-6-2(5x-3)}{5x-3}}{\frac{15x-10-3(5x-3)}{5x-3}}\). Then, simplify further: \(= \frac{9x-6-10x+6}{15x-10-15x+9} = \frac{9x-10x}{15x-15x}\) which simplifies to \(x\).

Step 4: verify the second condition

We also need to verify the second condition, \(f^{-1}(f(x))=x\). By replacing \(f(x)\) into \(f^{-1}(x)\), we get \(f^{-1}(f(x)) = f^{-1}\left(\frac{3x-2}{5x-3}\right)\). If we simplify as in step 3, we should also get \(x\) as result.

Step 5: Conclusion

Since both \(f(f^{-1}(x))=x\) and \(f^{-1}(f(x))=x\) hold, we can confirm that the function \(f(x)=\frac{3x-2}{5x-3}\) is indeed its own inverse.