Let's solve the problem step by step:

We are given:

• $f(x) = 3x + 3$
• $g(x) = 2 - 2x$

We need to find the value of $(f \circ g)^{-1}(5)$. Here's the step-by-step process:

Step 1: Compute $f \circ g(x)$

This is the composition of $f(x)$ and $g(x)$: $(f \circ g)(x) = f(g(x))$ Substitute $g(x) = 2 - 2x$ into $f(x)$: $f(g(x)) = f(2 - 2x)$ Substitute $f(x) = 3x + 3$: $f(2 - 2x) = 3(2 - 2x) + 3$ Simplify: $f(2 - 2x) = 6 - 6x + 3 = 9 - 6x$ Thus: $(f \circ g)(x) = 9 - 6x$

Step 2: Find the inverse of $f \circ g(x)$

We need $(f \circ g)^{-1}(y)$. Start by letting $y = f \circ g(x)$: $y = 9 - 6x$ Solve for $x$ in terms of $y$: $y - 9 = -6x$ $x = \frac{9 - y}{6}$ Thus: $(f \circ g)^{-1}(y) = \frac{9 - y}{6}$

Step 3: Compute $(f \circ g)^{-1}(5)$

Substitute $y = 5$ into the inverse: $(f \circ g)^{-1}(5) = \frac{9 - 5}{6}$ Simplify: $(f \circ g)^{-1}(5) = \frac{4}{6} = \frac{2}{3}$

Final Answer:

$(f \circ g)^{-1}(5) = \frac{2}{3}$

Do you want a detailed explanation of any step, or help with related problems? Here are five related questions you can explore:

1. How do you find the inverse of a composite function in general?
2. What is the domain of $f \circ g(x)$ for these specific functions?
3. How would the solution change if $f(x)$ or $g(x)$ were nonlinear functions?
4. Can you graph $f(x)$, $g(x)$, and $f \circ g(x)$ for better understanding?
5. How does the derivative of $f \circ g(x)$ relate to the derivatives of $f(x)$ and $g(x)$?

Tip: Always double-check the domain restrictions of functions when computing inverses