INVERSE FUNCTIONS
Prepared by: RAPHAEL V. PEREZ,
CpE
INVERSE FUNCTIONS
• In short, the reflector of the original
function at the radical axis y = x
• The original function is
...
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 ...
y = f(x)
(a1, b1)
(a2, b2)
(a3, b3)
(an+1, bn+1)
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...
INVERSE FUNCTIONS
EXAMPLE : FIND THE INVERSE
FUNCTION OF THE FOLLOWING:
f(x) = (-2,-6), (2,-4), (6,-2), (10,0)
Ans:
f-1(x)...
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,...
INVERSE FUNCTIONS
Now, in terms of POLYNOMIAL FUNCTION. Here are
the steps to get the inverse function [f-1(x)] of the
ori...
INVERSE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
When you get “x” on the composition method,
meaning our answer is correct.
GRAPH!
-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...
INVERSE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
POSSIBLE TO GRAPH ?
You may use the graphical software
for Cartesian and Polar coordinates
CLICK HERE
INVERSE FUNCTIONS
QUESTIONS?
For graphing: you will graph only linear functions ( y = mx + b).
Other functions like:
expon...
INVERSE FUNCTIONS
If you want this application
program for graphing purposes
install on your Personal
Computer,
visit www....
INVERSE FUNCTIONS
5
4
5xxf
25
4
5
1 x
xf
COMBINATION OF
OPERATIONS OF FUNCTIONS
Prepared by: RAPHAEL V.
PEREZ
RECALL: OPERATION OF
FUNCTIONS
EVALUATE THE FUNCTIONS
EVALUATE THE FUNCTIONS
EVALUATE THE FUNCTIONS
EVALUATE THE FUNCTIONS
EVALUATE THE FUNCTIONS
EVALUATE THE FUNCTIONS
EVALUATE THE FUNCTIONS
Inverse functions [repaired]
Inverse functions [repaired]
Inverse functions [repaired]
Inverse functions [repaired]
Inverse functions [repaired]
Inverse functions [repaired]
Inverse functions [repaired]
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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
  10. 10. INVERSE FUNCTIONS
  11. 11. INVERSE FUNCTIONS
  12. 12. INVERSE FUNCTIONS
  13. 13. INVERSE FUNCTIONS
  14. 14. INVERSE FUNCTIONS
  15. 15. INVERSE FUNCTIONS
  16. 16. INVERSE FUNCTIONS
  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)
  19. 19. INVERSE FUNCTIONS
  20. 20. INVERSE FUNCTIONS
  21. 21. INVERSE FUNCTIONS
  22. 22. INVERSE FUNCTIONS
  23. 23. INVERSE FUNCTIONS
  24. 24. INVERSE FUNCTIONS
  25. 25. INVERSE FUNCTIONS
  26. 26. INVERSE FUNCTIONS
  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
  35. 35. EVALUATE THE FUNCTIONS
  36. 36. EVALUATE THE FUNCTIONS
  37. 37. EVALUATE THE FUNCTIONS
  38. 38. EVALUATE THE FUNCTIONS
  39. 39. EVALUATE THE FUNCTIONS
  40. 40. EVALUATE THE FUNCTIONS
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