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# Inverse functions [repaired]

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### Inverse functions [repaired]

1. 1. INVERSE FUNCTIONS Prepared by: RAPHAEL V. PEREZ, CpE
2. 2. INVERSE FUNCTIONS • In short, the reflector of the original function at the radical axis y = x • The original function is f(x) and then the inverse function of f(x) is: f-1(x) or F(x) in other books
3. 3. INVERSE FUNCTIONS • In terms of ordered pairs, the inverse of f(x) = (a,b) is f-1(x) = F(x) (b,a) • In short, the inverse of the set: f(x) = (a1,b1), (a2,b2), (a3,b3),…, (an+1,bn+1) is f-1 (x) = F(x) = (b1, a1), (b2,a2), (b3,a3),…, (bn+1,an+1)
4. 4. y = f(x) (a1, b1) (a2, b2) (a3, b3) (an+1, bn+1)
5. 5. y = f(x) (a1, b1) (a2, b2) (a3, b3) (an+1, bn+1) (b1, a1) (b2, a2) (b3, a3) (bn+1, an+1) The inverse of f(x): f-1(x) = F(x) The set of ordered pairs at f(x) has been inverted
6. 6. INVERSE FUNCTIONS EXAMPLE : FIND THE INVERSE FUNCTION OF THE FOLLOWING: f(x) = (-2,-6), (2,-4), (6,-2), (10,0) Ans: f-1(x) = (-6,-2), (-4,2), (-2,6), (0,10)
7. 7. 8 -6 -4 -2 2 4 6 8 10 12 14 16 -6 -4 -2 2 4 6 8 y Axis y = x f(x) = (-2,-6), (2,-4), (6,-2), (10,0) f-1(x) = (-6,-2), (-4,2), (-2,6), (0,10) (-2,-6) (2,-4) (6,-2) (10,0) (-6,-2) (-4,2) (-2,6) (0,10)
8. 8. INVERSE FUNCTIONS Now, in terms of POLYNOMIAL FUNCTION. Here are the steps to get the inverse function [f-1(x)] of the original function f(x): 1. Change f(x) to “y” on the given function. 2. Invert the variables between x and y. The y variable in (1) will be “x” and for x variable on right side will be “y”. 3. Solve for y from (2). 4. Change “y” into f-1(x). 5. Solve for f [f-1(x)] and f-1[f(x)] (Composition Method). The answer must be “x”.
9. 9. INVERSE FUNCTIONS
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17. 17. INVERSE FUNCTIONS When you get “x” on the composition method, meaning our answer is correct. GRAPH!
18. 18. -5 -4 -3 -2 -1 1 2 3 4 5 6 -3 -2 -1 1 2 3 4 5 y INVERT THE ORDERED PAIRS FROM f(x) to graph (no need to solve) P1 P2 x 0 3 y 0 P1 P2 x 0 y 0 3 (3,0) (0, -3/2) (0,3) (-3/2, 0)
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27. 27. INVERSE FUNCTIONS POSSIBLE TO GRAPH ? You may use the graphical software for Cartesian and Polar coordinates CLICK HERE
28. 28. INVERSE FUNCTIONS QUESTIONS? For graphing: you will graph only linear functions ( y = mx + b). Other functions like: exponential (y = bx) logarithmic (y = logb x or y = ln x) , trigonometric (y = a sin x) and second degree or higher polynomials (y = axn + xn-1 +…+a0) are not yet discussed for way of sketching the function, sometimes you need to use programmable and graphical calculators or the computers. It’s hard to sketch the mentioned functions.
29. 29. INVERSE FUNCTIONS If you want this application program for graphing purposes install on your Personal Computer, visit www.padowan.dk this is a free-download software program.
30. 30. INVERSE FUNCTIONS
31. 31. 5 4 5xxf 25 4 5 1 x xf
32. 32. COMBINATION OF OPERATIONS OF FUNCTIONS Prepared by: RAPHAEL V. PEREZ
33. 33. RECALL: OPERATION OF FUNCTIONS
34. 34. EVALUATE THE FUNCTIONS
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36. 36. EVALUATE THE FUNCTIONS
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38. 38. EVALUATE THE FUNCTIONS
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40. 40. EVALUATE THE FUNCTIONS