7­7 Day 2 Inverse Relations and Functions March 04, 2009
7­7 Inverse Relations and Functions
Day 2
Objective: Find the inverse of a relation or function.
Feb 28­12:54 PM
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7­7 Day 2 Inverse Relations and Functions March 04, 2009
Graph each pair of functions on a single coordinate plane.
1.
3.
2.
Feb 28­12:55 PM
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7­7 Day 2 Inverse Relations and Functions March 04, 2009
Mar 4­9:14 AM
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7­7 Day 2 Inverse Relations and Functions March 04, 2009
The inverse of the function f is denoted by f ­1.
Read f ­1 as "the inverse of f " or as "f inverse."
Even if f(x) is a function, f ­1 may not be a function.
Feb 28­1:00 PM
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7­7 Day 2 Inverse Relations and Functions March 04, 2009
Example #1: Finding an Inverse Function
Given the function: f(x) = √x + 1
a. Find the domain and range of f.
D: x > ­1
R: y > 0 b. Find f ­1.
f(x) = √x + 1
y = √x + 1
Switch x and y. Then, solve for y.
x = √y + 1
Since x equals the principal square root, x > 0.
x2 = y + 1
y = x2 ­ 1
so f ­1(x) = x2 ­ 1, x > 0
Feb 28­1:00 PM
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7­7 Day 2 Inverse Relations and Functions March 04, 2009
Example #1: Finding an Inverse Function (continued)
Given the function: f(x) = √x + 1
Inverse: f ­1(x) = x2 ­ 1
c. Find the domain and range of f ­1.
D: x > 0
R: y > ­1
d. Is f ­1 a function? Explain.
Yes, f ­1 is a function. The graph f ­1 of passes the vertical line test.
Feb 28­1:01 PM
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7­7 Day 2 Inverse Relations and Functions March 04, 2009
Example #2: Finding an Inverse Function
Let f(x) = 10 ­ 3x
a. Find the domain and range of f.
b. Find f ­1.
c. Find the domain and range of f ­1.
Feb 28­1:01 PM
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7­7 Day 2 Inverse Relations and Functions March 04, 2009
Example #2: Finding an Inverse Function (continued)
Let f(x) = 10 ­ 3x
d. Find f ­1(f(3)).
and
x ­ 10
f ­1(x) = ­3
e. Find f(f ­1(2)).
Feb 28­1:01 PM
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7­7 Day 2 Inverse Relations and Functions March 04, 2009
Real­Life Problems:
The variables used in real­life problems are usually chosen specifically by letter. Do not interchange these variables to find an inverse, just solve for a different variable.
Feb 28­1:01 PM
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7­7 Day 2 Inverse Relations and Functions March 04, 2009
Example #3: Real­World Connection
r2
The function d = is a model for the distance d in feet that a 24
car with locked brakes skids in coming to a complete stop from a speed of r miles per hour. a. Find the inverse of the function. 2
d = r
24
r2 = 24d
r = √24d
b. What was the speed of a car that made skid marks 114 feet long rounded to the nearest mile per hour.
r = √24d
r = √24(114)
r ≈ 52 mph
Feb 28­1:01 PM
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7­7 Day 2 Inverse Relations and Functions March 04, 2009
Inverse Functions
If f and f ­1 are both functions, they are called inverse functions.
If f and f ­1 are both functions, and if f maps a to b, then f ­1 must map b to a.
Feb 28­1:02 PM
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7­7 Day 2 Inverse Relations and Functions March 04, 2009
Example #4: Composition of Inverse Functions
For the function f(x) = 5x + 11, find the following.
a. Find (f ­1 o f)(777)
­1
o
b. Find (f f
)(­5802)
Feb 28­1:02 PM
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7­7 Day 2 Inverse Relations and Functions March 04, 2009
Homework:
page 410
(23 ­ 34 all, 36 ­ 68 even)
Mar 1­12:42 PM
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