Question:

Step 1: We are given the relationship between two invertible functions. Given: $h(x) = 3g(x) + 7$ Step 2: To find the inverse function $h^{-1}(x)$, we use the definition that if $h(y) = x$, then $y = h^{-1}(x)$. Let us set $h(y) = x$ and substitute the expression for $h(x)$. $$3g(y) + 7 = x$$ Step 3: We solve for $g(y)$ by isolating it on one side of the equation. $$3g(y) = x - 7$$ $$g(y) = \frac{x-7}{3}$$ Step 4: Since $g$ is invertible, we apply the inverse function $g^{-1}$ to both sides of the equation. This allows us to solve for $y$. $$g^{-1}\left(g(y)\right) = g^{-1}\left(\frac{x-7}{3}\right)$$ $$y = g^{-1}\left(\frac{x-7}{3}\right)$$ Step 5: Since $y = h^{-1}(x)$ by our definition from Step 2, we have found the inverse function. $$h^{-1}(x) = g^{-1}\left(\frac{x-7}{3}\right)$$ **Final Answer:** The inverse function is $h^{-1}(x) = g^{-1}\left(\dfrac{x-7}{3}\right)$, which corresponds to **Option 4**.