Problem 106 - 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. $$\left(g^{-1} \circ f^{-1}\right)(-3)$$

Step-by-Step Solution

Answer

The value of \(\left(g^{-1} \circ f^{-1}\right)(-3)\) is 0.

Step 1: Find the inverse of \(f(x)\)

The function \(f(x)\) is given by \(f(x)=\frac{1}{8} x-3\). To find the inverse of the function \(f(x)\), first replace \(f(x)\) with \(y\), then swap \(x\) and \(y\) and solve for \(y\). In this case, if \(y=\frac{1}{8} x-3\), through algebraic steps, we find that the inverse function \(f^{-1}(x)\) is \(f^{-1}(x)=8x+24\).

Step 2: Evaluate \(f^{-1}(-3)\)

Plug -3 into the inverse function of \(f\), thus \(f^{-1}(-3)=8*(-3)+24=-24+24=0\). The result will be the input for the inverse of the function \(g\).

Step 3: Find the inverse of \(g(x)\)

The function \(g(x)\) is given by \(g(x)=x^{3}\). For this function, if \(y = x^3\), then the inverse function is obtained by rewriting it as \(x = y^3\) and solving for \(y\). Thus, \(g^{-1}(x)\) becomes \(g^{-1}(x)=\sqrt[3]{x}\).

Step 4: Evaluate \(g^{-1}(f^{-1}(-3))\)

The result from the second step which is 0, will be used as input for the inverse of \(g\). Therefore, \(g^{-1}(f^{-1}(-3)) = g^{-1}(0)= \sqrt[3]{0} = 0.\)