Question: Target 7: To evaluate a function for a given value, simply plug the value into the function for Evaluate each function for the given value. 67. f(x) = x² - 6x + 2 f(3) = -7 68. g(x) = 6x - 7 g(x+h) = (6x + 6h) - 7 69. f(x) = 3x² - 4 5[f(x+2)] = ______ Composition and Inverses of Functions: Recall: (f ∘ g)(x) = f[g(x)] OR f[g(x)] read “f of g of x” Means to plug the inside function (in this case g(x)) in for x in the outside function (in this case, f(x)). Example: Given f(x)=2x²+1 and g(x)=x-4 find f(g(x)). f(g(x)) = f(x-4) = 2(x-4)² + 1 = 2(x² - 8x + 16) + 1 = 2x² - 16x + 32 + 1 f(g(x)) = 2x² - 16x + 33 Suppose f(x) = 2x, g(x) = 3x - 2, and h(x) = x² - 4. Find the following: 70. f[g(2)] = 4 71. f[g(x)] = ______ 72. f[h(3)] = ______ 73. g[f(x)] = ______

1. **State the problem:** We are given functions and asked to evaluate or find compositions of functions for given values or expressions. 2. **Evaluate problem 67:** Given $$f(x) = x^2 - 6x + 2$$ and asked to find $$f(3)$$. 3. **Calculate $$f(3)$$:** $$f(3) = 3^2 - 6(3) + 2 = 9 - 18 + 2 = -7$$ 4. **Evaluate problem 68:** Given $$g(x) = 6x - 7$$, find $$g(x+h)$$. 5. **Calculate $$g(x+h)$$:** $$g(x+h) = 6(x+h) - 7 = 6x + 6h - 7$$ 6. **Evaluate problem 69:** Given $$f(x) = 3x^2 - 4$$, find $$5[f(x+2)]$$. 7. **Calculate $$f(x+2)$$:** $$f(x+2) = 3(x+2)^2 - 4 = 3(x^2 + 4x + 4) - 4 = 3x^2 + 12x + 12 - 4 = 3x^2 + 12x + 8$$ 8. **Calculate $$5[f(x+2)]$$:** $$5[f(x+2)] = 5(3x^2 + 12x + 8) = 15x^2 + 60x + 40$$ 9. **Recall composition of functions:** $$(f \circ g)(x) = f(g(x))$$ means plug $$g(x)$$ into $$f$$. 10. **Given:** $$f(x) = 2x$$, $$g(x) = 3x - 2$$, $$h(x) = x^2 - 4$$. 11. **Evaluate problem 70:** Find $$f[g(2)]$$. 12. **Calculate $$g(2)$$:** $$g(2) = 3(2) - 2 = 6 - 2 = 4$$ 13. **Calculate $$f[g(2)] = f(4)$$:** $$f(4) = 2(4) = 8$$ (Note: The user states $$f[g(2)] = 4$$ but calculation shows 8; we follow math.) 14. **Evaluate problem 71:** Find $$f[g(x)]$$. 15. **Calculate $$f[g(x)]$$:** $$g(x) = 3x - 2$$ $$f[g(x)] = f(3x - 2) = 2(3x - 2) = 6x - 4$$ 16. **Evaluate problem 72:** Find $$f[h(3)]$$. 17. **Calculate $$h(3)$$:** $$h(3) = 3^2 - 4 = 9 - 4 = 5$$ 18. **Calculate $$f[h(3)] = f(5)$$:** $$f(5) = 2(5) = 10$$ 19. **Evaluate problem 73:** Find $$g[f(x)]$$. 20. **Calculate $$f(x)$$:** $$f(x) = 2x$$ 21. **Calculate $$g[f(x)] = g(2x)$$:** $$g(2x) = 3(2x) - 2 = 6x - 2$$ **Final answers:** - 67. $$f(3) = -7$$ - 68. $$g(x+h) = 6x + 6h - 7$$ - 69. $$5[f(x+2)] = 15x^2 + 60x + 40$$ - 70. $$f[g(2)] = 8$$ - 71. $$f[g(x)] = 6x - 4$$ - 72. $$f[h(3)] = 10$$ - 73. $$g[f(x)] = 6x - 2$$