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Chapter 1: Problem 85

Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$\left(f^{-1} \circ f^{-1}\right)(6)$$

Short Answer

Expert verified
If we apply the function \((f^{-1} \circ f^{-1})(6)\) as indicated, the result will be 600.

Step by step solution

01

- Find the inverse of the function f

We first find the inverse of the function \(f(x)=\frac{1}{8} x-3\), which is denoted by \(f^{-1}(x)\). We start by replacing \(f(x)\) by \(y\) so our equation is: \(y=\frac{1}{8} x-3\). Then, swap \(x\) and \(y\): \(x=\frac{1}{8} y-3\). Solving this for \(y\), we get \(y=8x+24\). So, \(f^{-1}(x)=8x+24\).
02

- Apply the inverse function twice

As indicated by the notation \((f^{-1} \circ f^{-1})(6)\), we should apply the inverse function twice to the number 6. Starting out: \(f^{-1}(6) = 8*6 + 24 = 72\). Applying \(f^{-1}\) to the result again: \(f^{-1}(72) = 8*72 + 24 = 600\).
03

- Verify the result

Always remember to double-check your calculations. In this case, we can do so by applying the original function \(f(x)\) to the result twice. This should bring us back to our initial number. Checking: \(f(600) = \frac{1}{8} * 600 - 3 = 72 - 3 = 69\) and then \(f(69) = \frac{1}{8} * 69 - 3 = 5.625 - 3 = 2.625\). As you can see, it doesn't bring us back to our initial number. Thus, there seems to be a mistake.
04

- Review the calculations

Having noticed an error, we need to review our calculations. Looking back at 'Step 2', the mistake lies in the computation of the values. We have found the inverse function to be \(f^{-1}(x)=8x+24\). However, when we computed \(f^{-1}(6)\) we didn't use this formula correctly. The correct computation should be: \(f^{-1}(6)=8*6+24=72\), and then again \(f^{-1}(72)=8*72+24=600\).
05

- Correct the calculations and verify the result

Let's correct the error in our calculations from 'Step 2'. Recalculating: \(f^{-1}(6) = 8*6 + 24 = 72\) and then \(f^{-1}(72) = 8*72 + 24 = 600\). Now, let's verify this result by reapplying the original function \(f(x)\) twice to value 600. Calculating: \(f(600) = \frac{1}{8} * 600 - 3 = 72 - 3 = 69\) and then \(f(69) = \frac{1}{8} * 69 - 3 = 5.625 - 3 = 2.625\). This still doesn't bring us back to our initial number. Thus, our calculations seem off.
06

- Discover the error

Looking at 'Step 5', the mistake is in the verification step. While applying the original function twice, we have again used the formula incorrectly. The correct formula for \(f(x)\) is: \(f(x)=\frac{1}{8} x-3\). So, the computations should go as follows: Starting with 600, we should compute \(f(600) = \frac{1}{8} * 600 - 3 = 72\). And then, calculating \(f(72) = \frac{1}{8} * 72 - 3 = 6\). This brings us back to 6, our original number, and verifies the correctness of our previous steps.

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