Inverse Functions

1. Day 7 ­ Inverses
An inverse relation maps the output values back to their original input values.
f(x) f ­1(x)
1 6
2 7
3 8
4 9
5
10
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2. To find the inverse: change the coordinate pair (flip the x and y)
Example 1: g(x) = {(1, 5), (2, 6), (3, 7), (4, 8)}
g­1(x) =
Example 2: f (x) =
-3 4
2 6
3 9
f ­1(x)=
2

3. Example 3: Find the inverse of f(x) = 2x + 4
1. Switch x and y
2. Solve for y
Example 4: Find the inverse of h(x) = x2
Are the inverses functions? What is their domain? Range?
3

4. Example 5: Find the inverse of g(x) = (x ­ 2)2 + 1
Example 6: Find the inverse of j(x) = √ x + 5
Are the inverses functions? What is their domain? Range?
4

5. Example 7: Given f(x) = 2x + 6
a. Graph f(x)
b. Is f(x) a function?
c. Graph f ­1(x)
d. Is f ­1(x) a function?
Conclusion: A function and its inverse are reflections over y = x
5

6. Example 8:
a. Given g(x), sketch g ­1(x)
b. Find (g (g­1(2))
Conclusion: When you compose a function and it's inverse, you
always get what you started with...
6

7. From Ex 3: f(x) = 2x + 4 and f ­1(x) = ½ x ­ 2
From Ex 5: g(x) = (x ­ 2)2 + 1 and g ­1(x) = √x­1 + 2
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