Transcript Inverse Functions - crazymathteacher

Inverse Functions
Lesson
Warm-Up Included
Algebra I
Warm-Up for Inverse Function Lesson
REVIEW
1. A relation is a set of _______________ ___________.
2. A function is a special relation where there is ________________________________
_______________________________________________________________________.
3. Create ordered pairs for the function displayed in the mapping.
-6
-4
-1
0
6
-2
0
3
4
6
Talk about a regular function is
The inverse function is f ( x)
f ( x)
1
Really you are just switching your
input and output, or x and y values.
Show ordered pairs of
4. Inverse Operations UNDO one another. State the inverse operation.
a. Addition
b. Subtraction
c. Multiplication
d. Division
e. Squaring
f 1 ( x )
If you watched the Kahn Academy video, he talked
through solving for the opposite variable, solving for x
(which is highly unusual with a linear function … we
always solve for y … think of y=mx+b, etc.) Then he
showed swapping out the y for the x at the end.
That’s all fine and good, but I actually find it easier to
swap the variables in the first place and solve for y (like
we are used to doing).
If you understand that INVERSES undo and go
backwards … (instead of DR … RD; instead of
xy, yx) swap them right up front!
Suppose you are given the following directions:
•From home, go north on Rt 23 for 5 miles
•Turn east (right) onto Orchard Street
•Go to the 3rd traffic light and turn north (left) onto Avon Drive
•Tracy’s house is the 5th house on the right.
If you start from Tracy’s house, write down the directions to get home.
How did you come up with the directions to get home from Tracy’s?
Suppose you are given the following algorithm:
•Starting with a number, add 5 to it
•Divide the result by 3
•Subtract 4 from that quantity
•Double your result
The final result is 10. Working backwards knowing this result,
find the original number. Show your work.
Suppose you are given the following algorithm:
•Starting with a number, add 5 to it
•Divide the result by 3
•Subtract 4 from that quantity
•Double your result
The final result is 10. Working backwards knowing this result,
find the original number. Show your work.
Write a function f(x), which when given a number x (the original
number) will model the operations given above.
Write a function g(x), which when given a number x (the final
result), will model the backward algorithm that you came up with
above.
Find the algebraic inverse.
(Swap x and y, then solve for y.)
1.
2.
f ( x )  15 x  1
g ( x) 
1
x7
3
On the Kahn Academy video, you were shown how a
function and it’s inverse reflect over the line y=x..
This is a great example.
The green line is f ( x ) .
The red line is f 1 ( x ) .
On the Kahn Academy video, you were shown how a
function and it’s inverse reflect over the line y=x..
This is a CRAZY example.
The green line is f ( x ) .
The red line is f 1 ( x ) .
Is the inverse a function? If the inverse is not a function, how can you restrict
the domain of the original function so that the inverse is also a function?
Is the inverse a function? If the inverse is not a function, how can you restrict
the domain of the original function so that the inverse is also a function?
Go to Kahn Academy and watch the video for
Inverse Functions Example 2
https://www.khanacademy.org/math/algebra/algebrafunctions/function_inverses/v/function-inverses-example-2
1
2
f
(
x
)
f
(
x
)

(
x

2)
 1, for x  2
Find
if