Transcript Inverse Relations & Inverse Functions

Inverse Relations & Inverse
Functions
Lesson 1 (introduction)
S.E. 2A.4C: describe and analyze the
relationship between a function and its
inverse
Inverse Relations & Inverse Functions
Introduction
Real Life situation -concept of Inverse relation (Warm
up)
Suppose you are given the following directions:
• From home go left on Route 23 for 5 miles
• Turn right onto Orchard St.
• Turn left onto Avon Dr.
• Tracy’s house is the 5th house on the right.
If you start from Tracy’s house, write down the directions to get home:
•
•
•
•
From Tracy’s house, go left on Avon Dr.
Turn right on Orchard St.
Turn left on Route 23
You are home!
Inverse Relations & Inverse Functions
Guiding Question / Math application
• How did you come up with the directions to
get home from Tracy’s?
• ANSWER: Worked backward and reversed
directions and turns. This is how the concept
of “Inverse Relations and Inverse Functions”
works.
• Similarly, in Math you can reverse operations
to arrive at the inverse of a mathematical rule
or relation.
The inverse relation is the set of ordered pairs obtained by reversing
the coordinates of each ordered pair. This means that whenever f
contains (a,b) f-1(x) contains (b,a).
Determine the Inverse of a Function Defined by a
Map, a Table, or a Set of Ordered Pairs
PRACTICE
Find the inverse of the following function.
Finding the Inverse of a Function Defined by a Table
Find the inverse of the following function.
x
y
x
y
1
3
3
1
2
5
5
2
4
9
9
4
7
15
15
7
PRACTICE
Finding the Inverse of a Function Defined by a Table
Find the inverse of the following function.
x
y
-3
5
-2
9
-1
2
0
11
Find the inverse of the following function.
f(x) = {(-3, -27), (-2,-8), (-1,-1), (0,0), (1,1), (2,8), (3,27)}
f-1(x) = {(-27, -3), (-8,-2), (-1,-1), (0,0), (1,1), (8,2), (27,3)}
Find the Domain and Range of the Inverse
Function
Domain and Range
What is the domain of f(x)?
How does this relate to f-1(x)?
f-1(x)
f(x)
x
y
x
y
1
3
3
1
What is the range of f(x)?
How does this relate to f-1(x)?
2
5
5
2
The range of f is
the domain of f -1
4
9
9
4
7
15
15
7
The domain of f is
the range of f -1
PRACTICE
Finding the Domain and Range of an Inverse Function
The function f(x) is defined below.
f(x) = {(-2, 5), (-1,3), (3,7), (4,12)}
Find the range of f-1(x).
The function g(x) is defined below.
g(x) = {(6, 11), (-2,7), (0,3), (-5,4)}
Find the range of g-1(x).
Graphing with Patty Paper
• Graph y= 2x – 6 on graph paper
• Graph y=x on graph paper
• Trace y= 2x – 6, y=x, the x-axis and the y-axis on a
sheet of patty paper
• Fold the patty paper along the line y=x
• Trace the reflection of y= 2x – 6 onto the patty paper
• Trace the reflection of y= 2x – 6 onto the graph paper
using a different color pencil or make it
dotted/dashed.
Graphing with Patty Paper
1. What is the equation of the reflection (new line)?
2. List 3 points that lie on y= 2x – 6 and list the
coordinates of corresponding points on the
reflected graph.
3. What do you notice about the coordinates of the
corresponding points?
4. What are the domain and range of y= 2x – 6?
5. What are the domain and range of the reflection
of y= 2x – 6?
PRACTICE
The function f(x) is graphed below.
Graph the inverse function.
Which of the following is the graph of the function
below and its inverse?
A)
C)
B)
D)
Process for finding the Inverse Function
1. Switch x and y
2. Solve for y
Find the inverse of y  2 x
x  2y
x2  2 y
x2
y
2
1. Exchange x and y
2. Solve for y
Example
Find the inverse of the following function.
y=2x+5
1. Switch x and y.
x=2y+5
2. Solve for y.
x-5=2y (Subtract 5)
x-5
(x-5)/2=y (Divide by 2)
y=
2