1. **State the problem:** We are given three functions: $$f(x) = 3x, \quad g(x) = x^2, \quad h(x) = 2x - 5$$ We need to find the following composite functions: (a) $f(f(x))$ (b) $g(h(x))$ (c) $f(g(2x))$ (d) $h(g(x + 1))$ 2. **Recall the definition of composite functions:** For two functions $f$ and $g$, the composite function $f(g(x))$ means you first apply $g$ to $x$, then apply $f$ to the result. 3. **Calculate each part:** (a) $f(f(x)) = f(3x) = 3 \times (3x) = 9x$ (b) $g(h(x)) = g(2x - 5) = (2x - 5)^2$ Expand the square: $$ (2x - 5)^2 = (2x)^2 - 2 \times 2x \times 5 + 5^2 = 4x^2 - 20x + 25 $$ (c) $f(g(2x)) = f((2x)^2) = f(4x^2) = 3 \times 4x^2 = 12x^2$ (d) $h(g(x + 1)) = h((x + 1)^2) = 2 \times (x + 1)^2 - 5$ Expand $(x + 1)^2$: $$ (x + 1)^2 = x^2 + 2x + 1 $$ So: $$ h(g(x + 1)) = 2(x^2 + 2x + 1) - 5 = 2x^2 + 4x + 2 - 5 = 2x^2 + 4x - 3 $$ 4. **Final answers:** (a) $f(f(x)) = 9x$ (b) $g(h(x)) = 4x^2 - 20x + 25$ (c) $f(g(2x)) = 12x^2$ (d) $h(g(x + 1)) = 2x^2 + 4x - 3$ These results show how to combine functions step-by-step by substitution and simplification.