Section 1.6. Inverse Functions

Transcription

1 Section 1.6 Inverse Functions

2 Important Vocabulary Inverse function: Let f and g be two functions. If f(g(x)) = x in the domain of g and g(f(x) = x for every x in the domain of f, then g is the inverse function of the function f. The function is denoted by f 1. One-to-one: A function f one-to-one if, for a and b in its domain, f(a) = f(b) implies a = b. Horizontal Line Test: A function is one-to-one if every horizontal line intersects the graph of the function at most once.

3 I. Inverse Functions For a function f that is defined by a set of ordered pairs, to form the inverse function of f,... interchange the first and second coordinates of each these ordered pairs. For a function f and its inverse f 1, the domain of f is equal to the range of f 1, and the range of f is equal to the domain of f 1

4 I. Inverse Functions To verify that two functions, f and g, are inverses of each other,... find f(g(x)) and g(f(x)). If both of these compositions are equal to the identity function (x), then the functions are inverses of each other.

5 Example 1: Verify that the functions f(x) = 2x 3 and are inverses of each other.

6 Example 1: Verify that the functions f(x) = 2x 3 and are inverses of each other. f(g(x)) = 2( ) 3 = x = x g(f(x) = Therefore the functions are inverses of each other.

7 II. The Graph of an Inverse Function If the point (a, b) lies on the graph of f, then the point ( b, a ) lies on the graph of f 1 and vice versa. The graph of f 1 is a reflection of the graph of f in the line y = x. Numerically verify that f(x) and g(x), from Example 1, are inverses. Graphically verify that f(x) and g(x) from are inverses. Use the viewing window Xmin = -8, Xmax = 8, Ymin = -6, Ymax = 6

8 Example 1: Numerically verify that the functions f(x) = 2x 3 and each other. are inverses of f(x) g(x) x y x y Since for each point (a, b) on the graph of f, the point (b, a) lies on g(x), these functions are inverses.

9 Example 1: Graphically verify that the functions f(x) = 2x 3 and each other. are inverses of

10 III. The Existence of an Inverse Function A function f has an inverse f 1 if and only if... f is one-to-one. If a function is one-to-one, that means... that no two elements in the domain of the function corresponds to the same element in the range of the function To tell whether a function is one-to-one from its graph,... simply use the Horizontal Line Test, that is, check to see that every horizontal line intersects the graph of the function at most one.

11 Example 2: Does the graph of the function at the right have an inverse function? Explain.

12 Example 2: Does the graph of the function at the right have an inverse function? Explain. No, it doesn t pass the Horizontal Line Test.

13 IV. Finding Inverse Functions Algebraically To find the inverse of a function f algebraically,... 1) Use the Horizontal Line Test to decide whether f has an inverse function. 2) In the equation for f(x), replace f(x) by y. 3) Interchange the roles of x and y, and solve for y. 4) Replace y by f 1 (x) in the new equation. 5) Verify that f and f 1 are inverse functions of each other by showing that the domain of f is equal to the range of f 1, the range of f is equal to the domain of f 1, and f(f 1 (x)) = x and f 1 (f(x)) = x

14 Example 3: Find the inverse (if it exists) of f(x) = 4x 5.

15 Example 3: Find the inverse (if it exists) of f(x) = 4x 5. Use the Horizontal Line Test to decide whether f has an inverse function. f passes the Horizontal Line Test so it will have an inverse function.

16 Example 3: Now find the inverse of f(x) = 4x 5. y = 4x 5 replace f(x) by y. x = 4y - 5 interchange the roles of x and y x + 5 = 4y solve for y f 1 (x) = 0.25x replace y by f 1 (x) in the new equation Continued...

17 Example 3: Show that the domain of f is equal to the range of f 1, the range of f is equal to the domain of f 1 f(x) = 4x 5, Domain: (, ) f 1 (x) = 0.25x , Range: (, ) These are equal f(x) = 4x 5, Range: (, ) f 1 (x) = 0.25x , Domain: (, ) These are equal Continued...

18 Example 3: Lastly, we must show that f(f 1 (x)) = x and f 1 (f(x)) = x f(x) = 4x 5 f 1 (x) = 0.25x f(f 1 (x)) = f(0.25x ) = 4(0.25x ) 5 = x = x f 1 (f(x)) = f 1 (4x 5) = 0.25(4x 5) = x = x Therefore, f 1 (x) = 0.25x is confirmed.

19 Example 4: Find the inverse function of f(x) = x - 6 informally. Then verify that both f(f 1 (x)) and f 1 (f(x)) are equal.

20 Example 4: Find the inverse function of f(x) = x 6 informally. Then verify that both f(f 1 (x)) and f 1 (f(x)) are equal. Solution: The function f subtracts 6 from each input. To undo this function, you need to add 6 to each input. So, the inverse function of f(x) = x 6 is given by f 1 (x) = x + 6 f(f 1 (x)) = f(x + 6) = (x + 6) 6 = x f 1 (f(x)) = f 1 (x 6) = (x 6) + 6 = x