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Lesson 7-1 Polynomial Functions Lesson 7-2 Graphing Polynomial Functions Lesson 7-3 Solving Equations Using Quadratic Techniques Lesson 7-4 The Remainder and Factor Theorems Lesson 7-5 Roots and Zeros Lesson 7-6 Rational Zero Theorem Lesson 7-7 Operations on Functions Lesson 7-8 Inverse Functions and Relations Lesson 7-9 Square Root Functions and Inequalities Contents

Example 1 Find an Inverse Relation Example 2 Find an Inverse Function Example 3 Verify Two Functions are Inverses Lesson 8 Contents

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 and determine whether the resulting ordered pairs are also the coordinates of the vertices of a rectangle. 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 8-1a

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 8-1b

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. Find the inverse of this relation and determine whether the resulting ordered pairs are also the coordinates of the vertices of a pentagon. Answer: {(4, –3), (5, –1), (3, 2), (1, 1), (1, –2)} These ordered pairs also describe the vertices of a pentagon. Example 8-1c

Step 1 Replace f (x) with y in the original equation. Find the inverse of Step 1 Replace f (x) with y in the original equation. Step 2 Interchange x and y. Example 8-2a

Step 4 Replace y with f –1(x). Step 3 Solve for y. Inverse Multiply each side by –2. Add 2 to each side. Step 4 Replace y with f –1(x). Example 8-2b

Answer: The inverse of is Example 8-2c

Graph the function and its inverse. Graph both functions on the coordinate plane. The graph of is the reflection for over the line Example 8-2d

Answer: Example 8-2e

b. Graph the function and its inverse. Answer: a. Find the inverse of b. Graph the function and its inverse. Answer: Answer: Example 8-2f

Determine whether and are inverse functions. Check to see if the compositions of f (x) and g (x) are identity functions. Example 8-3a

Answer: The functions are inverses since both and equal x. Example 8-3b

Determine whether and are inverse functions. Answer: The functions are inverses since both compositions equal x. Example 8-3c