Function Composition 6B5709

1. **State the problem:** We are given two functions: $$f(x) = 3x^2 + 2x$$ $$g(x) = 5x - 6$$ We need to find the composite functions $(f \circ g)(x)$ and $(g \circ f)(x)$ and simplify them completely. --- 2. **Recall the definition of composite functions:** - $(f \circ g)(x) = f(g(x))$ means we substitute $g(x)$ into $f(x)$. - $(g \circ f)(x) = g(f(x))$ means we substitute $f(x)$ into $g(x)$. --- 3. **Find $(f \circ g)(x)$:** Substitute $g(x) = 5x - 6$ into $f(x)$: $$ (f \circ g)(x) = f(g(x)) = f(5x - 6) = 3(5x - 6)^2 + 2(5x - 6) $$ Expand the square: $$ (5x - 6)^2 = (5x)^2 - 2 \times 5x \times 6 + 6^2 = 25x^2 - 60x + 36 $$ Substitute back: $$ (f \circ g)(x) = 3(25x^2 - 60x + 36) + 2(5x - 6) $$ Distribute: $$ = 75x^2 - 180x + 108 + 10x - 12 $$ Combine like terms: $$ = 75x^2 - 170x + 96 $$ --- 4. **Find $(g \circ f)(x)$:** Substitute $f(x) = 3x^2 + 2x$ into $g(x)$: $$ (g \circ f)(x) = g(f(x)) = g(3x^2 + 2x) = 5(3x^2 + 2x) - 6 $$ Distribute: $$ = 15x^2 + 10x - 6 $$ --- 5. **Final answers:** $$ (f \circ g)(x) = 75x^2 - 170x + 96 $$ $$ (g \circ f)(x) = 15x^2 + 10x - 6 $$