Five-Minute Check (over Lesson 4–6)
CCSS
Then/Now
New Vocabulary
Key Concept: Inverse Relations
Example 1: Inverse Relations
Example 2: Graph Inverse Relations
Key Concept: Finding Inverse Functions
Example 3: Find Inverse Linear Functions
Example 4: Real-World Example: Use an Inverse Function
Over Lesson 4–6
Which equation of a regression line best represents
the table? Let x be the number of years since 2001.
A. y = 4.3x + 15.2
B. y = 5.7x + 16.1
C. y = 2x + 13
D. y = x + 3
Over Lesson 4–6
Which equation of a regression line best represents
the table?
A. y = 25x + 13.2
B. y = 0.08x + 14.3
C. y = 0.16x + 14.61
D. y = 0.25x + 12
Over Lesson 4–6
Which equation of a regression line best represents
the table?
A. y = 200.2x – 30
B. y = 190x – 20
C. y = 97x + 7
D. y = 54x – 18
Over Lesson 4–6
A line of best fit for a set of data has slope 6.5 and
passes through the point at (–8, 40). What is the
regression equation?
A. y = x + 70
B. y = 4x + 40
C. y = 5.5x + 92
D. y = 6.5x + 92
Content Standards
A.CED.2 Create equations in two or more
variables to represent relationships between
quantities; graph equations on coordinate axes
with labels and scales.
F.BF.4a Solve an equation of the form f (x ) = c
for a simple function f that has an inverse and
write an expression for the inverse.
Mathematical Practices
6 Attend to precision.
Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State
School Officers. All rights reserved.
You represented relations as tables, graphs,
and mappings.
• Find the inverse of a relation.
• Find the inverse of a linear function.
• inverse relation – the set of ordered pairs
obtained by exchanging the x-coordinates
with the y-coordinates of each ordered pair
in a relation
• inverse function – can generate ordered
pairs of the inverse relation
Notice that the domain of a relation becomes
the range of its inverse, and the
range of the relation becomes the domain
of its inverse.
Find the inverse of each relation.
1) {(4, -10), (7, -19), (-5, 17), (-3, 11)}
To find the inverse, exchange the
coordinates of the ordered pairs
(4, -10)
(-10, 4)
(7, -19)
(-19, 7)
(-5, 17)
(17, -5)
(-3, 11)
(11, -3)
The inverse is {(-10, 4), (-19, 7), (17, -5),
(11, -3)}
x
-4
-1
5
9
y
-13
-8.5
0.5
6.5
Write the coordinates as ordered pairs.
• Then exchange the coordinates of each
pair.
• (-4, -13)
(-13, -4)
• (-1, -8.5)
(-8.5, -1)
• (5, 0.5)
(0.5, 5)
• (9, 6.5)
(6.5, 9)
• The inverse is {(-13, -4), (-8.5, -1), (0.5, 5),
(6.5, 9)}
• Find the inverse of each relation.
x
y
-6
8
-15
11
9
3
0
6
Write the ordered pairs
{(-6, 8), (-15, 11), (9, 3), (0, 6)}
Exchange the coordinates
The inverse is {(8, -6), (11, -15), (3, 9), (6, 0)}
Inverse Relations
A. Find the inverse of each relation.
{(−3, 26), (2, 11), (6, −1), (−1, 20)}
To find the inverse, exchange the coordinates of the
ordered pairs.
(–3, 26) → (26, –3)
(6, –1) → (–1, 6)
(2, 11) → (11, 2)
(1, 20) → (20, 1)
Answer: The inverse is {(26, –3), (11, 2), (–1, 6),
(20, –1)}.
Inverse Relations
B. Find the inverse of each relation .
Write the coordinates as ordered pairs.
Then exchange the coordinates of
each pair.
(4, 3) → (3, 4)
(–2, 0) → (0, –2)
(1, 4.5) → (4.5, 1)
(5, 10.5) → (10.5, 5)
Answer: The inverse is {(3, 4), (4.5, 1), (0, –2),
(10.5, 5)}.
Find the inverse of {(4, 8), (–6, 6), (3, 3), (0, –8)}.
A. {(4, 8), (–6, 6), (3, 3), (0, –8)}
B. {(8, 4), (6, –6), (3, 3), (–8, 0)}
C. {(0, –8), (3, 3), (–6, 6), (4, 8)}
D. {(–4, –8), (6, –6), (–3, –3), (0, 8)}
Graph Inverse Relations
A. Graph the inverse of each relation.
Graph Inverse Relations
Answer: The graph of the relation passes through
the points at (–2, 6), (2, 0), and (6, 6). To find points
through which the graph of the inverse passes,
exchange the coordinates of the ordered pairs. The
graph of the inverse passes through the points at
(6, –2), (0, 2), and (6, 6). Graph these points and
then draw the line that passes through them.
Graph Inverse Relations
B. Graph the inverse of each relation.
Graph Inverse Relations
Answer: The graph of the relation passes through
the points at (–2,– 6), (0, 4), (2, 0), (4, –4), and
(6, –8). To find points through which the graph of
the inverse passes, exchange the coordinates of
the ordered pairs. The graph of the inverse passes
through the points at (-6, -2), (4, 0), (0, 2), (–4, 4),
and (–8, 6). Graph these points and then draw the
line that passes through them.
Graph the inverse of the relation.
A.
B.
C.
D.
Find Inverse Linear Functions
A. Find the inverse of the function f (x) = –3x + 27.
Step 1
f(x) = –3x + 27
Original equation
y = –3x + 27
Replace f(x) with y.
Step 2
x = –3y + 27
Interchange y and x.
Step 3
x – 27 = –3y
Subtract 27 from each side.
x – 27 = y
3 3
Divide each side by –3.
Find Inverse Linear Functions
Simplify.
Step 4
Answer: The inverse of f(x) = –3x + 27 is
• Find the inverse of each function.
f(x) = 4x – 8
Replace f(x) with y
y = 4x – 8
Interchange y and x
x = 4y – 8
Solve for y
x = 4y – 8
+8
+8
x + 8 = 4y
4 4 4
x/4 + 2 = y
Replace y with f(x)-1
f(x)-1 = 1/4x + 2
The inverse of f)x) = 4x – 8 is f(x)-1 = 1/4x + 2
When graphed on the same coordinate plane
f(x)-1 appears to be the reflection of f(x)
• f(x) = 4x – 12
y = 4x – 12
x = 4y – 12
+12
+ 12
x + 12 = 4y
4
4 4
y = 1/4x + 3
f(x)-1 = 1/4x + 3
Find Inverse Linear Functions
Step 1
Original equation
Replace f(x) with y.
Step 2
Interchange y and x.
Step 3
Add 8 to each side.
Find Inverse Linear Functions
Simplify.
Step 4
Answer:
f(x) = - ½ x + 11
y = - ½ x + 11
x = - ½ y + 11
-11
-11
x – 11 = - ½ y
-2(x – 11) = - ½ y(-2)
-2x + 22 = y
f(x)-1 = -2x + 22
f(x) = 1/3x + 7
y = 1/3x + 7
x = 1/3y + 7
-7
-7
x – 7 = 1/3y
3(x – 7) = 1/3y(3)
3x – 21 = y
f(x)-1 = 3x - 21
Find the inverse of f(x) = 12 – 9x.
A.
B.
C.
D.
• Find the inverse of each function
1) f(x) = 17 – 1/3x
f(x)-1 = -3x + 51
2) f(x) = 12 – 6x
f(x)-1 = -1/6x + 2
3) f(x) = -16 – 4/3x
f(x)-1 = -3/4x – 12
4) f(x) = 5/11x + 10
f(x)-1 = 11/5x – 22
5) f(x) = -2/3x + ¼
f(x)-1 = -3/2x + 3/8
Use an Inverse Function
SALES Carter sells paper supplies and makes a
base salary of $2200 each month. He also earns 5%
commission on his total sales. His total earnings
f(x) for a month in which he compiled x dollars in
total sales is f(x) = 2200 + 0.05x.
A. Find the inverse function.
Step 1
f(x) = 2200 + 0.05x
y = 2200 + 0.05x
Step 2
x = 2200 + 0.05y
Original equation
Replace f(x) with y.
Interchange y and x.
Use an Inverse Function
Step 3
x – 2200 = 0.05y
Subtract 2200 from each side.
Divide each side by 0.05.
Step 4
Answer:
Use an Inverse Function
SALES Carter sells paper supplies and makes a
base salary of $2200 each month. He also earns 5%
commission on his total sales. His total earnings
f(x) for a month in which he compiled x dollars in
total sales is f(x) = 2200 + 0.05x.
B. What do x and f –1(x)represent in the context of
the inverse function?
Answer: x represents Carter’s total earnings for the
month and f –1(x) represents the total monthly sales by
Carter for the company.
Use an Inverse Function
SALES Carter sells paper supplies and makes a
base salary of $2200 each month. He also earns 5%
commission on his total sales. His total earnings
f(x) for a month in which he compiled x dollars in
total sales is f(x) = 2200 + 0.05x.
C. Find Carter’s total sales for last month if his
earnings for that month were $3450.
Carter earned $3450 for the month. To find Carter’s total
sales for that month, find f –1(3450).
Use an Inverse Function
f –1(x) = 20x – 44,000
Original equation
f –1(3450) = 20(3450) – 44,000
total earnings = $3450
f –1(3450) = 69,000 – 44,000
Multiply.
f –1(3450) = 25,000
Subtract.
Answer: Carter had $25,000 in total sales for the
month.
• Randall is writing a report on Santiago, Chile,
and he wants to include a brief climate
analysis. He found a table of temperatures
recorded in degrees Celsius. He knows that a
formula for converting degrees Fahrenheit to
degrees Celsius is C(x) = 5/9(x – 32). He will
need to find the inverse function to convert
from degrees Celsius to Fahrenheit.
Average Temp. (0C)
Month
Min
Max
Jan
12
29
Mar
9
27
May
5
18
July
3
15
Sept
6
29
Nov
9
26
Find the inverse function C-1(x)
C(x) = 5/9(x – 32)
• C(x) = 5/9(x – 32)
y = 5/9(x – 32)
x = 5/9(y – 32)
x = 5/9y – 160/9
9/5x = 9/5(5/9y – 32)
9/5x = y – 32
+32
+32
9/5x + 32 = y
C-1(x) = 9/5x + 32
• What do x and C-1(x) represent in the
context of the inverse function?
x represents the temperature in degrees
Celsius and C-1(x)n represents the
temperature in degrees Fahrenheit.
• Find the average temperatures for July in
degrees Fahrenheit.
The average min is 30C and max is 150C
Find the average min and max for Fahrenheit
C-1(x) = 9/5x + 32
C-1(3) = 9/5(3) + 32
= 37.4
C-1(15) = 9/5(15) + 32
= 59
The average temp min is 37.40F and max id
590F for July
• Peggy rents a car for the day. The total cost
C(x) in dollars is given by C(x) = 19.99 + 0.3x,
where x is the number of miles she drives.
1) Find the inverse function
C-1(x) = 1/3x – 19.99/3
2) What do x and C-1(x) represent in the context
of the inverse function?
x is the total cost and C-1(x) is the total number
of miles driven
3) How many miles did Peggy drive if her total
cost was $34.99?
50 miles
REPAIRS Nikki’s car is getting repairs. The mechanic is
charging her $40 to look at the car and $65 for each
half-hour to fix the car. Her total cost f(x) for the repairs
is f(x) = 40 + 65x. Find the inverse function and how
long it took the mechanic to fix the car if Nikki was
charged a total of $365.
A.
B.
C.
D.