Inverse Functions

1
Section 1.7

• Inverse Functions

2
Bad Humor of the day
Two brooms were hanging in the closet and after
a while they got to know each other so well, they
decided to get married. One broom was, of course,
the bride broom, the other the groom broom. The
bride broom looked very beautiful in her white
dress. The groom broom was handsome and suave in
his tuxedo. The wedding was lovely. After the
wedding, at the wedding dinner, the bride-broom
leaned over and said to the groom-broom, "I think
I am going to have a little whisk
broom!!!" "IMPOSSIBLE !!" said the groom
broom. "WE HAVEN'T EVEN SWEPT TOGETHER!"
3
Section 1.7
Suppose we are given two functions f(x) and g(x).
then f(x) and g(x) are inverses of each other
Example
To prove two functions are inverses, you must
find the two compositions!!!
4
Section 1.7
Notation If f(x) and g(x) are found to be
inverses of each other, then g(x) f-1(x)
f-1 is read as f - inverse
When a function is proven to have an inverse, the
following characteristics will hold true The
domain of f(x) the range of f-1(x) The domain
of f-1(x) the range of f(x)
5
Section 1.7
For a single function to have an inverse, the
function must be one-to-one.
We can determine if a function is one-to-one by
using the horizontal line test.
Example f(x) x3
Notice that any horizontal line drawn across the
graph of f(x) x3 intersects at one and only
one point
F(x) x3 is a one-to-one function
6
Section 1.7
If any single horizontal line intersects the
graph more than once anywhere, the graph is not
one-to-one
Example f(x) x2 - 2
Notice that any horizontal line drawn across the
graph of f(x) x2 - 2 intersects twice above
the vertex (0, 2)
F(x) x2 - 2 is NOT a one-to-one function
7
Section 1.7
If a function has been proven to have an inverse,
we can find the inverse using the following steps
1. Change the notation from f(x) to y.
2. In the equation, exchange the xs and the ys
3. Solve the equation for y.
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Section 1.7
Example
9
Section 1.7

• Assignment 9
• Read Pg. 171 - 176
• Problems Pg. 177 - 179
• 1, 3, 13 - 23 (part a only) odd,
• No Graphs?39 - 57 odd, 67, 69