Problem 110
Question
Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$g^{-1} \circ f^{-1}$$
Step-by-Step Solution
Verified Answer
The composite function \(g^{-1} \circ f^{-1}\) is \(\sqrt[3]{8x+24}\)
1Step 1: Find the Inverse of f
To find the inverse of a function, you should switch \(y\) (or \(f(x)\)) with \(x\). Here, \(f(x)=\frac{1}{8}x-3\) becomes \(x = \frac{1}{8}y - 3\). Solving for \(y\) gives \(f^{-1}(x) = 8x + 24\).
2Step 2: Find the Inverse of g
Applying the same process to the function \(g\), we get \(x = y^{3}\). To get the inverse function, we solve for \(y\), giving \(y = \sqrt[3]{x}\). So \(g^{-1}(x) = \sqrt[3]{x}\).
3Step 3: Find the Composite Function
Now we need to form the composite function \(g^{-1} \circ f^{-1}\) by substituting \(f^{-1}(x)\) into \(g^{-1}(x)\). We get \(g^{-1}(f^{-1}(x))= \sqrt[3]{8x+24}\).
Other exercises in this chapter
Problem 109
Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$f^{-1} \circ g^{-1}$$
View solution Problem 109
Think About It Let \(f\) be an even function. Determine whether \(g\) is even, odd, or neither. Explain. (a) \(g(x)=-f(x)\) (b) \(g(x)=f(-x)\) (c) \(g(x)=f(x)-2
View solution Problem 111
Prove that a function of the following form is odd. $$y=a_{2 n+1} x^{2 n+1}+a_{2 n-1} x^{2 n-1}+\cdots+a_{3} x^{3}+a_{1} x$$
View solution Problem 111
Use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$g^{-1} \circ f^{-1}$$
View solution