1. INVERSE FUNCTION
Given a function y = f(x), when we solve for x in terms of y , we get the equation x = g(y). If for each y there exists only one x paired with it then x = g(y) defines a new function which will be known as the inverse of f, denoted by f -1(x). This will happen only when y = f(x) is a one to one function.
Thus, if f is one to one then it will have an inverse f -1 .
Terms of y
Interchange x and y
The graph of f and f -1 are symmetrical to line y = x. 1

2. A function y = f(x) is monotonic if and only if it is 1 – 1 .
A function y = f(x) is said to be monotonic on an interval I if it is either strictly increasing or strictly decreasing on that interval.
A function y = f(x) is monotonic if and only if it is 1 – 1 . x y
V(h,k)
NOT monotonic NOT 1-1
Monotonic
Terms of y
Horizontal Line Test for 1-1 Functions
If any horizontal line intersects the graph of f at exactly one point, then f is 1-1. 2

3. F = {(1, a), (2, b), (3, d), (4, c)}
DF : { 1, 2, 3, 4 }
Since no two ordered pairs have the same second components, F is 1-1 so F has an inverse
RF: { a, b, d, c }
DF-1 : {a, b, d, c }
RF-1 : {1, 2, 3, 4 }

5. Solving for x in terms of y
y = f(x) = 3x +3
y = 3x+3
Solving for x in terms of y
y = x
y = (1/3)x-1
Interchange x and y x -1
f(x) 3 x 3
f -1(x) -1
Df : R
Df-1 : R
Rf : R
Rf-1 : R 5

6. Solving for x in terms of y
NOT monotonic so NOT 1-1
For each y there correspond 2 x’s
so the equation does not define a function. g has NO inverse.
Restrict domain to make it monotonic so it will be 1-1 and therefore has an inverse. 6

7. Solving for x in terms of y
y = x
Interchange x and y
Dg :
Dg-1 : x -2
g(x) 1 5 x 1 5
g -1(x) -2
Rg :
Rg-1 : 7

8. -2 1 5 1 5 -2 Dg : Dg-1 : Rg : Rg-1 : x g(x) x g-1(x) (-2, 5) y = x-2
g(x) 1 5
(-2, 5)
y = x x 1 5
g-1(x) -2
(0, 1)
(1, 0)
Dg :
(5,-2)
Dg-1 :
Rg :
Rg-1 : 8

9. Solving for x in terms of y
y = x
Interchange x and y
Dg :
Dg-1 :
Rg :
Rg-1 : 9

10. 2 1 5 1 5 2 Dg : Dg-1 : Rg : Rg-1 : x g(x) x (2, 5) g-1(x) y = x2
g(x) 1 5
(2, 5)
y = x
(5, 2) x 1 5
g-1(x) 2
(0, 1)
(1, 0)
Dg :
Dg-1 :
Rg :
Rg-1 : 10

11. Solving for x in terms of y
y = x
Interchange x and y
Dg :
Dg-1 :
Rg :
Rg-1 : 11

12. -2 7 3 3 -2 7 Dg : Dg-1 : Rg : Rg-1 : x g(x) x (3, 7) g-1(x) y = x3
(3, 7)
y = x x 3
g-1(x) -2 7
(7, 3)
(-2, 0)
Dg :
Dg-1 :
(0, -2)
Rg :
Rg-1 : 12

13. Solving for x in terms of y
(-3,6)
Solving for x in terms of y
y = 1
(0, 3)
(-7, 2)
(3,0)
(6,-3)
y = -2
x = -2
(2,-7)
x = 1
Dh: all x R except x = -2
Dh-1: all x R except x = 1
Rh: all y R except y = 1
Rh-1: all y R except y = -2 13

14. 3 -3 -7 6 2 x 6 2 3 -3 -7 x h(x) h-1 (x) y = 1 y = -2 x = -2 x = 13 -3 -7
h(x) 6 2
(-3,6)
y = 1
(-7, 2)
(0, 3) x 6 2
h-1 (x) 3 -3 -7
(3,0)
(6,-3)
x = -2
y = -2
x = 1
(2,-7) 14

15. PATIENCE
AND
HARD WORK
ARE THE KEYS TO
DO GOOD IN MATH