Questions: f(x)=x+7 g(x)=3x-2 Find (f ∘ g)^-1(x)

f(x)=x+7 g(x)=3x-2 Find (f ∘ g)^-1(x)
Transcript text: $\begin{array}{l}f(x)=x+7 \quad g(x)=3 x-2 \\ \text { Find }(f \circ g)^{-1}(x)\end{array}$
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Solution

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Solution Steps

To find \((f \circ g)^{-1}(x)\), we need to follow these steps:

  1. Compute the composition \(f(g(x))\).
  2. Find the inverse of the resulting function.
Step 1: Define the Functions

Given the functions: \[ f(x) = x + 7 \] \[ g(x) = 3x - 2 \]

Step 2: Compute the Composition \( f(g(x)) \)

To find the composition \( f(g(x)) \): \[ f(g(x)) = f(3x - 2) = (3x - 2) + 7 = 3x + 5 \]

Step 3: Find the Inverse of the Composition

To find the inverse of the function \( f(g(x)) = 3x + 5 \), we solve for \( x \) in terms of \( y \): \[ y = 3x + 5 \] \[ y - 5 = 3x \] \[ x = \frac{y - 5}{3} \]

Thus, the inverse function is: \[ (f \circ g)^{-1}(x) = \frac{x - 5}{3} \]

Final Answer

\(\boxed{(f \circ g)^{-1}(x) = \frac{x - 5}{3}}\)

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