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

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

Using f(x) = 3x + 2 and g(x) = 2x2 – 1, find (f – g)(x), if it exists.

Over Lesson 6–1

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

Using f(x) = 3x + 2 and g(x) = 2x2 – 1, find (f ● g)(x), if it exists.

Over Lesson 6–1

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

Using f(x) = 3x + 2 and g(x) = 2x2 – 1, find [f ○ g](x), if it exists.

Over Lesson 6–1

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

Using f(x) = 3x + 2 and g(x) = 2x2 – 1, find [g ○ f](x), if it exists.

Over Lesson 6–1

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

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.

Over Lesson 6–1

A. f(1)

B. g(1)

C. (g ○ f)(1)

D. f(0)

Let f(x) = x – 3 and g(x) = x2. Which of the following is equivalent to (f ○ g)(1)?

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.

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)}

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?

Find and Graph an Inverse

Step 1 Replace f(x) with y in the original equation.

Then graph the

function and its inverse.

Step 2 Interchange x and y.

Find and Graph an Inverse

Step 3 Solve for y.

Inverse

Step 4 Replace y with f –1(x).

y = –2x + 2 f –1(x) = –2x + 2

Multiply each side by –2.

Add 2 to each side.

Find and Graph an Inverse

Find and Graph an Inverse

Answer:

A.

B.

C.

D.

Graph the function

and its inverse.

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.