Part 4 Function Operations Composition & Inverses

IV. Function Operations, Compositions, and Inverses (Ch. 1-841-9) I PERFORM THE FUNCTION OPERATIONS & COMPOSITIONS: consider functions f and g: Given f and g, find: X f(x) X g(x) 0 5 0 3 2 2 2 -2 3 3 3 0 5 -1 5 5 a. f(3)=3 f. (f/g)(5) undefined b. g(3)=0 g. (5/7)(3):0 c. (f+g)(0)> h. (3 i. (gf) (0) 5 -1 d. (f-g)(5)=-6 e. (fg)(2)=-4 consider the following functions: f(x) x²-4 and (g(x)=x+2) Given f and g, find: a. (f°g)(x) b. ((f+g)(x)·x²+x-2) c. (\frac{f}{9}(x)) [(f+g)(x)·x²+x-2] [(f_{0})(x)=(x²-η)(x+2)] [\frac{f}{9}(x)·\frac{x²-4}{x+2}] [(fg)(x)=x³+2x²-4x-8] [\frac{f}{9}(x)=x-2,x≠-2] d. (\frac{9}{f}(x)) e. ( f. (gof)(x) [(g°f)(x)=x²-4+2] [(g²°f)(x)=x²-2] [(f°a)(x)∈(y+2)²-4] [(f+0)(x)=x²+40|x+4-4] [(f+g)(x)=x²+4x] [\frac{9}{6}(x)+\frac{x+2}{x^{x}-4}] [\frac{9}{f}(x)-\frac{1}{x-2},x≠±2] consider functions: (h(x)=x+3) and (g(x)=x²) find (h°g)(x)= (x²+3) [(g°h)(x)=(x+3)] [=x²+6x+9] decompose the following composition: for (h(x)=\sqrt[2]{2x+1}) find two functions f and g: h(x)= (f°g)(x). g(x)= 2x1 f(x)=x(x) II. DETERMINE IF A FUNCTION IS 1-1 one to one function - each element of the range associates with ONLY one element of the range ordered pairs 6. table 7. graph HLT \[f(x)=\{(0,-1),(1,3),(4,-9),(7,2)\}\] X \[\underline{f(x)}\] 2 5 3 5 7 5 10 5 YES or NO YES or NO YES or NO III FUNCTIONS AND THEIR INVERSES: relationship between a function and it's inverse: \[g(x)\equiv f^{-1}(x)\Leftrightarrow f(g(x))\equiv g(f(x))\equiv x\] 8. If \(f(x)=\{(0,-1),(1,3),(4,-q),(1,2)\}\) find \(f^{-1}(x)\) \[f^{-1}(x)=\{(-9,4),(-1,0),(2,7),(3,1)\}\] 11. given f(x), graph \(f^{-1}(x)\) (5,7) 9. given \(f(x)=\frac{x+4}{2x-5}\) find f(x) \[x=\frac{w+4}{2y-5}\] \[2xy-5x=y+4\] \[2xy-y+5x+4\] \[y(2x-1)=5x+4\] \[f^{-1}(x)=\frac{5x+4}{2x-1}\] (0,4) \[y=f(x)\] (1,5) (3,1) \[y=f^{-1}(x)\] (4,0) (4.2) (1,3) \[y=x\] \[(-2,-4)\] 10. determine whether \(g(x)\equiv f^{-1}(x)\) given \(f(x)=\frac{3}{4}x-6\) and \(g(x)=\frac{4}{3}x+8\) \[f(g(x))=\frac{3}{9}(\frac{u}{5}x+b)-6\] \[x+6-6\] X \[g(x)\equiv f^{-1}(x)\] QED