Math 110
Section 4.2
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Inverse, 1-to-1 functions (§4.2):
Inverse of a function
If a given function, f(x), gets you some answer y (given some starting point x)
Then the inverse function, f 1  y  , takes the answer, and gets you the original starting point x
So, if f(x) converts from American dollars into British pounds, then f
pounds into American dollars
1
 y  converts British
Draw the circle pictures:
Students
(the domain)
GPA
(the range)
x
(the domain)
f(x)
(the range)
-7
-4
-1
2
5
8
11
-4
-1
2 of the arrow, and
Pictorially, we’ll invert the function by removing the arrow-heads from one end
putting it on the other end.
5
8
ICE: Table/Pairs: reverse X, Y
Bob
Jo
Mary
Joe
Sarah
Rob
Jane
2.0
3.8
4.0
3.9
3.5
2.9
2.1
-3
-2
-1
0
1
2
3
11
One-to-one Functions
Each input (each x value) produces (corresponds to) exactly one output (y value) (all functions do
this)
AND
Each output (each y value) produces (corresponds to) exactly one intput (x value)
(this makes it into a 1-to-1 fnx)
Passes both a vertical line AND a horizontal line test.
Symmetric about the y = x line
Draw a Couple of these on the board, and ask them about which are which, and why
Only 1-to-1 functions have inverse functions
Non 1-to-1 functions could be thought of as having inverse
relations, which are different
Math 110
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Math 110
Section 4.2
Page 2 / 2
For a given f(x), here's how to get the inverse function:
1. Replace f(x) with y
At the end of this step, we'll have y = blah x blah
2. Since we usually write f 1 x  instead of f 1  y  , we want to swap the x and y's
This means we'll have x = blah y blah, where x shows up once, and y may be occur a bunch
of times
3. Solve for y
At the end of this step, we're back at y = blah x blah
4. Replace y with f 1 x 
Example:
Step
f x   4 x  2
y  4x  2
(1)
x  4y  2
(2)
x  2  4y
(3)
x2
y
(3)
4
x2
f 1  x  
(4)
4


=
=
=
13.1.4 – Inverse, Composite Functions
Point out that if f(x) converts from American dollars into British pounds, then f 1 x  converts
British pounds into American dollars, therefore f f 1 x   x
This is the equivalent of converting from American dollars into
British pounds, then converting from British pounds back into
American dollars
Thus, for any function f(x), we can show that
and
f  f 1 x   x
f 1  f x   x






ICE: Do a couple of these
Math 110
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