Inverse Functions 5.3 Chapter 5 Functions 5.3.1 MATHPOWERTM 11, WESTERN EDITION 5.3.1

The inverse of a function is a reflection of its graph in Inverse Functions The inverse of a function is a reflection of its graph in the line y = x. It can be determined by interchanging the coordinates of the ordered pairs in the functions. The inverse of a function is written as f-1(x) and read as “the inverse of f at x.” When x and y are interchanged in the equation of a function: The coordinates of the points that satisfy the equation are interchanged. The graph of the function is reflected in the line y = x. To determine the inverse of a function: Interchange x and y in the equation of the function. Solve the resulting equation for y. 5.3.2

Graphing the Inverse Function Note: If the ordered pair (3, 6) satisfies the function f(x), then the ordered pair (6, 3) will satisfy the inverse, f-1(x). Find the inverse of the function f(x) = 4x - 7. y = 4x - 7 Interchange the x and y values. x = 4y - 7 x + 7 = 4y x + 7 = y 4 f-1(x) (-3, 1) (-7, 0) y = x f(x) (1, -3) (0, -7) 5.3.3

If two functions f(x) and g(x) are inverses of Verifying an Inverse If two functions f(x) and g(x) are inverses of each other, then f(g(x)) must equal x and g(f(x)) must equal x. Verify that the functions f(x) = 4x - 7 and are inverses. f(g(x)) must be equal to x. g(f(x)) must be equal to x. f(x) = 4x - 7 g(4x - 7) = (x + 7) - 7 = x = x Since f(g(x)) and g(f(x)) are both equal to x, then f(x) and g(x) are inverses of each other. 5.3.4

y = x Graphing a Function and Its Inverse Graph f(x) = x2 + 1 (-2, 5) (2, 5) The graphs are symmetrical about the line y = x. (1, 2) For the function: (-1, 2) (5, 2) Domain: Range: (2 , 1) y > 1 (0, 1) (1, 0) For the inverse: Domain: Range: x > 1 (2, - 1) y = x (5, -2) Is the inverse a function? Could the domain of f(x) be restricted so that the inverse is a function? 5.3.5

Graphing a Function and Its Inverse [cont’d] Given f(x) = x2 + 1, the inverse is NOT a function. Graph y = x2 + 1 where x > 0. Graph the inverse. Is this a function? Graph y = x2 + 1 where x < 0. Graph its inverse Is this a function? What are your conclusions about restricting the domain so that the inverse is a function? Vertical Line Test 5.3.6

Assignment Questions: Pages p. 99 #3, 5(a,c), 6(a,c,e), 10 and p. 105 #1(a,c), 2(a,c), 4, 6 - 10 5.3.7