Five-Minute Check (over Lesson 6–1)
CCSS
Then/Now
New Vocabulary
Key Concept: Inverse Relations
Example 1: Find an Inverse Relation
Key Concept: Property of Inverses
Example 2: Find and Graph an Inverse
Key Concept: Inverse Functions
Example 3: Verify that Two Functions are Inverses
Over Lesson 6–1
Using f(x) = 3x + 2 and g(x) = 2x2 – 1,
find (f – g)(x), if it exists.
A. (f – g)(x) = 2x2 + 3x + 1
B. (f – g)(x) = 2x2 + 3x + 3
C. (f – g)(x) = –x2 – x – 1
D. (f – g)(x) = –2x2 + 3x + 3
Over Lesson 6–1
Using f(x) = 3x + 2 and g(x) = 2x2 – 1,
find (f ● g)(x), if it exists.
A. (f ● g)(x) = 6x3 + 4x2 – 3x – 2
B. (f ● g)(x) = 6x2 + 4x – 2
C. (f ● g)(x) = 5x2 + 4x – 3
D. (f ● g)(x) = x2 + 6x + 1
Over Lesson 6–1
Using f(x) = 3x + 2 and g(x) = 2x2 – 1,
find [f ○ g](x), if it exists.
A. [f ○ g](x) = 12x2 + 6x + 1
B. [f ○ g](x) = 6x2 + 4x – 2
C. [f ○ g](x) = 6x2 – 1
D. [f ○ g](x) = 3x2 – 2
Over Lesson 6–1
Using f(x) = 3x + 2 and g(x) = 2x2 – 1,
find [g ○ f](x), if it exists.
A. [g ○ f](x) = 18x3 + 24x2 + 7x + 1
B. [g ○ f](x) = 18x2 + 24x + 7
C. [g ○ f](x) = 6x2 + 24x + 7
D. [g ○ f](x) = 6x3 + 4x2 – 3x – 2
Over Lesson 6–1
To obtain a retail price, a dress shop adds $20 to
the wholesale cost x of every dress. When the shop
has a sale, every dress is sold for 75% of the retail
price. If f(x) = x + 20 and g(x) = 0.75x, find [g ○ f](x)
to describe this situation.
A.
[g ○ f](x) = 0.75x + 15
B.
[g ○ f](x) = 0.75(x + 20) + 15
C.
[g ○ f](x) = x + 20(0.75x + 15)
D.
[g ○ f](x) = 1.75x + 35
Over Lesson 6–1
Let f(x) = x – 3 and g(x) = x2. Which of the following
is equivalent to (f ○ g)(1)?
A. f(1)
B. g(1)
C. (g ○ f)(1)
D. f(0)
Content Standards
F.IF.4 For a function that models a relationship between
two quantities, interpret key features of graphs and tables
in terms of the quantities, and sketch graphs showing key
features given a verbal description of the relationship.
F.BF.4.a Find inverse functions. - 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
7 Look for and make use of structure.
8 Look for and express regularity in repeated reasoning.
You transformed and solved equations for a
specific variable.
• Find the inverse of a function or relation.
• Determine whether two functions or relations
are inverses.
• inverse relation
• inverse function
Find an Inverse Relation
GEOMETRY The ordered pairs of the relation {(1, 3),
(6, 3), (6, 0), (1, 0)} are the coordinates of the vertices
of a rectangle. Find the inverse of this relation.
Describe the graph of the inverse.
To find the inverse of this relation, reverse the coordinates
of the ordered pairs. The inverse of the relation is {(3, 1),
(3, 6), (0, 6), (0, 1)}.
Find an Inverse Relation
Answer: Plotting the points shows that the ordered
pairs also describe the vertices of a rectangle.
Notice that the graph of the relation and the
inverse are reflections over the graph of y = x.
GEOMETRY The ordered pairs of the relation {(–3, 4),
(–1, 5), (2, 3), (1, 1), (–2, 1)} are the coordinates of the
vertices of a pentagon. What is the inverse of this
relation?
A. cannot be determined
B. {(–3, 4), (–1, 5), (2, 3), (1, –2)}
C. {(–3, 4), (–1, 5), (2, 3), (1, 1), (–2, 1)}
D. {(4, –3), (5, –1), (3, 2), (1, 1), (1, –2)}
Find and Graph an Inverse
Then graph the
function and its inverse.
Step 1
Replace f(x) with y in the original equation.
Step 2
Interchange x and y.
Find and Graph an Inverse
Step 3
Solve for y.
Inverse
Multiply each side by –2.
Add 2 to each side.
Step 4
Replace y with f –1(x).
y = –2x + 2  f –1(x) = –2x + 2
Find and Graph an Inverse
Find and Graph an Inverse
Answer:
Graph the function
and its inverse.
A.
B.
C.
D.
Verify that Two Functions are Inverses
Check to see if the compositions of f(x) and g(x) are
identity functions.
Verify that Two Functions are Inverses
Answer: The functions are inverses since both [f ○ g](x)
and [g ○ f](x) equal x.
A. They are not inverses since
[f ○ g](x) = x + 1.
B. They are not inverses since both
compositions equal x.
C. They are inverses since both
compositions equal x.
D. They are inverses since both
compositions equal x + 1.