Question

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

Short answer

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

Step-by-step solution

Find Inverse Function of \(f(x)\)

To find the inverse of the function \(f(x) = \frac{1}{8}x - 3\), swap \(x\) and \(y\), giving you \(x = \frac{1}{8}y - 3\). Then isolate \(y\) to have \(f^{-1}(x)\). That becomes \(f^{-1}(x) = 8x + 24\).

Evaluate \(f^{-1}(-3)\)

We substitute -3 into the inverse function of \(f(x)\), i.e. \(f^{-1}(-3) = 8(-3) + 24 = -24 + 24 = 0\)

Find Inverse Function of \(g(x)\)

To find the inverse of \(g(x) = x^{3}\), we swap \(x\) and \(y\), giving us \(x = y^{3}\). We then isolate \(y\) to get \(g^{-1}(x)\), resulting in \(g^{-1}(x) = \sqrt[3]{x}\).

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

Now that we have \(f^{-1}(-3) = 0\), we substitute this value into \(g^{-1}(x) = \sqrt[3]{x}\) which gives \(g^{-1}(f^{-1}(-3)) = \sqrt[3]{0} = 0\).