Questions: Use the functions given by f(x) = 1/8 x - 3 and g(x) = x^3 to find (f^(-1) o g^(-1))(8).

Use the functions given by f(x) = 1/8 x - 3 and g(x) = x^3 to find (f^(-1) o g^(-1))(8).
Transcript text: Use the functions given by $f(x)=\frac{1}{8} x-3$ and $g(x)=x^{3}$ to find $\left(f^{-1} \circ g^{-1}\right)(8)$.
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Solution

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Solution Steps

To solve for \((f^{-1} \circ g^{-1})(8)\), we need to find the inverse functions \(f^{-1}(x)\) and \(g^{-1}(x)\) first. Then, we apply \(g^{-1}\) to 8 and use the result as the input for \(f^{-1}\).

  1. Find \(f^{-1}(x)\): Solve the equation \(y = \frac{1}{8}x - 3\) for \(x\) in terms of \(y\).
  2. Find \(g^{-1}(x)\): Solve the equation \(y = x^3\) for \(x\) in terms of \(y\).
  3. Compute \(g^{-1}(8)\): Use the inverse function of \(g\) to find the value.
  4. Compute \(f^{-1}(g^{-1}(8))\): Use the result from step 3 as the input for \(f^{-1}\).
Step 1: Find the Inverse of \( f(x) \)

Given the function \( f(x) = \frac{1}{8}x - 3 \), we need to find its inverse. To do this, we solve the equation \( y = \frac{1}{8}x - 3 \) for \( x \):

\[ y = \frac{1}{8}x - 3 \implies x = 8y + 24 \]

Thus, the inverse function is \( f^{-1}(x) = 8x + 24 \).

Step 2: Find the Inverse of \( g(x) \)

Given the function \( g(x) = x^3 \), we find its inverse by solving \( y = x^3 \) for \( x \):

\[ y = x^3 \implies x = y^{1/3} \]

Thus, the inverse function is \( g^{-1}(x) = x^{1/3} \).

Step 3: Compute \( g^{-1}(8) \)

Using the inverse function \( g^{-1}(x) = x^{1/3} \), we compute:

\[ g^{-1}(8) = 8^{1/3} = 2 \]

Step 4: Compute \( f^{-1}(g^{-1}(8)) \)

Now, we use the result from Step 3 as the input for \( f^{-1}(x) = 8x + 24 \):

\[ f^{-1}(2) = 8 \times 2 + 24 = 40 \]

Final Answer

\(\boxed{40}\)

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