1. **Problem statement:** Given the functions $f(x) = 2x + 3$ and $g(x) = x^2$, calculate: (i) $f(f(x))$ (ii) $g(f(x))$ (iii) $f^{-1}(x)$ (the inverse of $f$) 2. **Formulas and rules:** - Composition of functions: $(f \, o \, g)(x) = f(g(x))$ - To find $f(f(x))$, substitute $f(x)$ into $f$. - To find $g(f(x))$, substitute $f(x)$ into $g$. - To find the inverse $f^{-1}(x)$, solve $y = f(x)$ for $x$ in terms of $y$. 3. **Calculations:** (i) Calculate $f(f(x))$: $$f(f(x)) = f(2x + 3) = 2(2x + 3) + 3 = 4x + 6 + 3 = 4x + 9$$ (ii) Calculate $g(f(x))$: $$g(f(x)) = g(2x + 3) = (2x + 3)^2 = (2x)^2 + 2 \times 2x \times 3 + 3^2 = 4x^2 + 12x + 9$$ (iii) Find the inverse $f^{-1}(x)$: Start with $y = 2x + 3$ Solve for $x$: $$y = 2x + 3 \implies 2x = y - 3 \implies x = \frac{y - 3}{2}$$ So, $$f^{-1}(x) = \frac{x - 3}{2}$$ **Final answers:** (i) $f(f(x)) = 4x + 9$ (ii) $g(f(x)) = 4x^2 + 12x + 9$ (iii) $f^{-1}(x) = \frac{x - 3}{2}$