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Max used these steps to find the inverse of function $f$.




Step Number		FunctionStep Description

Step 1		$f(x)=\frac{5}{6} x-\frac{1}{6}$given

Step 2		$y=\frac{5}{6} x-\frac{1}{6}$change $f(x)$ to $y$

Step 3		$x=\frac{5}{6} y-\frac{1}{6}$switch $x$ and $y$

Step 4		$x+\frac{1}{6}=\frac{5}{6} y$add $\frac{1}{6}$ to each side

Step 5		$\frac{5}{6} x+\frac{5}{36}=y$multiply each side by $\frac{5}{6}$

Step 6		$\frac{5}{6} x+\frac{5}{36}=f^{-1}(x)$replace $y$ with $f^{-1}(x)$






Use this information to complete the statement.

Max made a mistake in [Choose an answer], he should have [Choose an answer].

Verified Answer :

  • Analyze Max's steps to find the inverse of \(f(x) = \frac{5}{6}x - \frac{1}{6}\).
  • Identify that Step 5 contains an error where the wrong multiplier (\(\frac{5}{6}\) instead of \(\frac{6}{5}\)) is used to solve for \(y\).
  • Calculate the correct inverse by multiplying the equation \(x + \frac{1}{6} = \frac{5}{6}y\) by \(\frac{6}{5}\) to obtain \(y = \frac{6}{5}x + \frac{1}{5}\).
  • State the final correction: Max made a mistake in \(\boxed{\text{Step 5}}\), he should have \(\boxed{\text{multiplied each side by } \frac{6}{5}}\).

Explanation

  1. Problem Analysis

The objective is to find the inverse of the linear function \(f(x) = \frac{5}{6}x - \frac{1}{6}\). Max has provided a series of steps to compute this inverse. To determine the correct approach, we analyze each of his steps to ensure algebraic accuracy. The process for finding an inverse of \(y = f(x)\) generally involves swapping \(x\) and \(y\) and solving for the new \(y\). We must verify if each step correctly follows these logical operations.

  1. Identifying the Error

We examine Max's work starting from Step 4, where the equation is \(x + \frac{1}{6} = \frac{5}{6}y\). To isolate \(y\), one must eliminate the coefficient \(\frac{5}{6}\). Multiplying a variable by \(\frac{5}{6}\) is reversed by multiplying both sides of the equation by its multiplicative inverse (reciprocal), which is \(\frac{6}{5}\). Max, however, multiplied both sides by \(\frac{5}{6}\) instead. Specifically, in Step 5, he performed \(\frac{5}{6} \times (x + \frac{1}{6}) = \frac{5}{6} \times \frac{5}{6}y\), which yields \(\frac{5}{6}x + \frac{5}{36} = \frac{25}{36}y\), but he claimed this resulted in \(y\). Therefore, the mistake occurred in Step 5.

  1. Correcting the Operation

To correctly solve for \(y\) from the equation \(x + \frac{1}{6} = \frac{5}{6}y\), we multiply both sides by the reciprocal \(\frac{6}{5}\):
\[\frac{6}{5} \left( x + \frac{1}{6} \right) = \frac{6}{5} \left( \frac{5}{6}y \right)\]
Distributing the \(\frac{6}{5}\) on the left side:
\[\frac{6}{5}x + \left( \frac{6}{5} \times \frac{1}{6} \right) = y\]
\[\frac{6}{5}x + \frac{1}{5} = y\]
Thus, the correct inverse function is \(f^{-1}(x) = \frac{6}{5}x + \frac{1}{5}\). Max's multiplication by \(\frac{5}{6}\) was the incorrect operation.

  1. Final Conclusion

Based on our analysis, Max made a mistake in Step 5, where he should have multiplied both sides of the equation by \(\frac{6}{5}\). The final statement is: Max made a mistake in \(\boxed{\text{Step 5}}\), he should have \(\boxed{\text{multiplied each side by } \frac{6}{5}}\).

Examples

In many real-world scenarios, such as currency conversion, we often need to reverse a calculation. For example, if a function \(f(x) = 1.2x\) converts dollars to euros (where \(x\) is dollars and the rate is 1.2), to find the reverse function (euros to dollars), we must multiply by the reciprocal, \(\frac{1}{1.2} \approx 0.833\), rather than just multiplying by the same rate again. This ensures we correctly undo the initial transformation, just as we did to find the inverse of the linear function here.

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