1. **State the problem:** We have a function $g(x) = 2x - 5$ and need to find its inverse function $g^{-1}(x)$. 2. **Find the inverse function:** To find $g^{-1}(x)$, start by setting $y = g(x) = 2x - 5$. 3. Swap $x$ and $y$ to find the inverse: $$x = 2y - 5$$ 4. Solve for $y$: $$x + 5 = 2y$$ 5. Divide both sides by 2: $$y = \frac{x + 5}{2}$$ 6. Write the inverse function: $$g^{-1}(x) = \frac{x + 5}{2}$$ 7. **Show that $g^{-1}g(x) = gg^{-1}(x)$:** - Compute $g^{-1}(g(x))$: $$g^{-1}(g(x)) = g^{-1}(2x - 5) = \frac{(2x - 5) + 5}{2} = \frac{2x}{2} = x$$ - Compute $g(g^{-1}(x))$: $$g(g^{-1}(x)) = g\left(\frac{x + 5}{2}\right) = 2 \times \frac{x + 5}{2} - 5 = (x + 5) - 5 = x$$ 8. Since both compositions equal $x$ for every $x$, the equality $g^{-1}g(x) = gg^{-1}(x)$ holds. **Final answers:** $$g^{-1}(x) = \frac{x + 5}{2}$$ $$g^{-1}g(x) = gg^{-1}(x) = x$$