Chapter 3: Transformations of Graphs and Data Lesson 8: Inverse Functions Mrs. Parziale

Vocabulary If you switch the coordinates in a function, the resulting relation is called the inverse of the function. Given the relation {(1, 2), (3, 4), (5, 6), (7, 8)} The inverse is {(2,1), (4, 3), (6, 5), (8, 7)}

Example 1: Let g = { (-1, 1) (2, 4) (3, 9) (4, 16) } (a) Is g a function (check to see if the x’s repeat)? (b)Find the inverse of g: (switch all the x's and y's in each pair) (c) Is the inverse a function? _________ Why? ___________

Example 2: Find the equation for the inverse of f(x) = 4x - 1 (Switch x and y and solve for y). Graph both. What is the line of symmetry? _________________

Example 3: Using f(x) in Example 2, (a)Find f(f -1 (x)) (evaluate f at its inverse): (b) Find f -1 (f(x)) =

Inverse Functions Theorem: Two functions f and g are inverse functions if and only if: (a)For all x in the domain of f, g(f(x)) = x AND (b) For all x in the domain of g, f(g(x)) = x Since the results of the composites of inverses is the same, only these composites are commutative.

HORIZONTAL LINE TEST: This test is only for the inverse of a function. The inverse of a function f is also a function if and only if no horizontal line intersects the graph of f (original function) in more than one point.

Example 4: Graph the following, find an equation for the inverse, graph, determine if it is a function: Function? _______ Inverse a function? ____ Inverse: _________

Example 4: Graph the following, find an equation for the inverse, graph, determine if it is a function: Function? _______ Inverse a function? ____ Inverse: _________

Example 5: Find the inverse.

Example 6: Let Are f and g inverses? Justify your answer – find compositions in both directions.

Closure - Example 7: Let Find