Problem 105 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The value of \((f^{-1} \circ g^{-1})(1)\) is 32.

Step 1: Find the inverse of function \(g\)

Function \(g\) is given as \(g(x)=x^3\). To find its inverse, replace \(g(x)\) by \(y\): \(y = x^3\). The next step is to swap \(x\) and \(y\): \(x = y^3\). Finally, to solve for \(y\), take the cubed root of both sides: \(y = \sqrt[3]{x}\). So, \(g^{-1}(x) = \sqrt[3]{x}\).

Step 2: Evaluate \(g^{-1}(1)\)

Now, we have to evaluate \(g^{-1}(1)\), or substitute \(1\) into \(g^{-1}(x)\): \(g^{-1}(1) = \sqrt[3]{1}\). The cubed root of 1 is 1, so \(g^{-1}(1) = 1\).

Step 3: Find the inverse of function \(f\)

Function \(f\) is given as \(f(x)=\frac{1}{8} x-3\). To find its inverse, again, replace \(f(x)\) with \(y\): \(y = \frac{1}{8} x-3\). After swapping \(x\) and \(y\), we get \(x = \frac{1}{8} y-3\). Solving for \(y\) gives us \(y = 8x + 24\). So, \(f^{-1}(x) = 8x + 24\).

Step 4: Evaluate \(f^{-1}(g^{-1}(1))\)

Finally, we evaluate \(f^{-1}(g^{-1}(1))\) by substituting the value of \(g^{-1}(1)\) into \(f^{-1}(x)\): \(f^{-1}(1) = 8*1 + 24 = 32\).