1. **State the problem:** We are given two functions: $$g(x) = \frac{2x + 1}{5}$$ and $$f(x) = x + 4$$ We need to: - Calculate the value of $$g(-2)$$. - Write an expression for the composition $$g(f(x))$$ in simplest form. - Find the inverse function $$g^{-1}(x)$$. 2. **Calculate $$g(-2)$$:** Substitute $$x = -2$$ into $$g(x)$$: $$g(-2) = \frac{2(-2) + 1}{5} = \frac{-4 + 1}{5} = \frac{-3}{5}$$ 3. **Find $$g(f(x))$$:** The composition $$g(f(x))$$ means substituting $$f(x)$$ into $$g$$: $$g(f(x)) = g(x + 4) = \frac{2(x + 4) + 1}{5}$$ Simplify the numerator: $$2(x + 4) + 1 = 2x + 8 + 1 = 2x + 9$$ So, $$g(f(x)) = \frac{2x + 9}{5}$$ 4. **Find the inverse function $$g^{-1}(x)$$:** Start with: $$y = \frac{2x + 1}{5}$$ Swap $$x$$ and $$y$$ to find the inverse: $$x = \frac{2y + 1}{5}$$ Multiply both sides by 5: $$5x = 2y + 1$$ Isolate $$y$$: $$2y = 5x - 1$$ Divide both sides by 2: $$y = \frac{5x - 1}{2}$$ Therefore, $$g^{-1}(x) = \frac{5x - 1}{2}$$