Splash Screen

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 5-Minute Check 4

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. [g ○ f](x) = 0.75x + 15 [g ○ f](x) = 0.75(x + 20) + 15 C. [g ○ f](x) = x + 20(0.75x + 15) D. [g ○ f](x) = 1.75x + 35 5-Minute Check 5

Mathematical Practices 7 Look for and make use of structure. 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. CCSS

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. Then/Now

Definition of Inverse:

Concept

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)}. Example 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. Example 1

Concept

Then graph the function and its inverse. Find and Graph an Inverse Then graph the function and its inverse. Example 2

Find and Graph an Inverse Answer: Example 2

When the inverse of a function is a function. Horizontal Line Test: Determine if the inverse is a function. One-to-one Function: When the inverse of a function is a function. Function: For every input there is one output. Vertical line test: Determines if a graph is a function.

Find and Graph an Inverse Answer: Example 2

Concept

Verify that Two Functions are Inverses Example 3

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. Example 3

End of the Lesson