3.2 Inverse Functions. Functions A function maps each element in the domain to exactly 1 element in the range.

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Presentation transcript:

3.2 Inverse Functions

Functions A function maps each element in the domain to exactly 1 element in the range.

Concept 1

Example 1 Domain and Range State the domain and range of the relation. Then determine whether the relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. The relation is {(1, 2), (3, 3), (0, –2), (–4, –2), (–3, 1)}. Answer: The domain is {–4, –3, 0, 1, 3}. The range is {–2, 1, 2, 3}. Each member of the domain is paired with one member of the range, so this relation is a function. It is onto, but not one-to-one.

Concept 2

Which of the following are functions?

Concept

Example 1 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.

Concept

Example 2 Find and Graph an Inverse Step 1Replace f(x) with y in the original equation. Then graph the function and its inverse. Step 2Interchange x and y.

Example 2 Find and Graph an Inverse Step 3Solve for y. Step 4Replace y with f –1 (x). y = –2x + 2  f –1 (x) = –2x + 2

STOP!! Finish on Monday

Example 2 A. B. C. D. Graph the function and its inverse.

Example 3: Find the inverse of f(x)

Concept

Example 4 Verify that Two Functions are Inverses Check to see if the compositions of f(x) and g(x) are identity functions.

Example 4 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.