Inverse Functions

Objectives  Students will be able to find inverse functions and verify that two functions are inverse functions of each other.  Students will be able to use the graph of a functions to determine whether functions have inverse functions.  Students will be able to use the horizontal line test to determine if functions are one- to-one.  Find inverse functions algebraically.

Definition of the Inverse Function Let f and g be two functions such that f (g(x)) = x for every x in the domain of g and g(f (x)) = x for every x in the domain of f. The function g is the inverse of the function f, and denoted by f -1 (read “f-inverse”). Thus, f ( f -1 (x)) = x and f -1 ( f (x)) = x. The domain of f is equal to the range of f -1, and vice versa.

Example Show that each function is the inverse of the other: f (x) = 3x and g(x) = x/3. Solution To show that f and g are inverses of each other, we must show that f (g(x)) = x and g( f (x)) = x. We begin with f (g(x)). f (x) = 3x f (g(x)) = 3g(x) = 3(x/3) = x. Next, we find g(f (x)). g(x) = x/3 g(f (x)) = f (x)/3 = 3x/3 = x. Notice how f -1 undoes the change produced by f.

Graphing Inverses Inverses are symmetric about the line y=x To graph reverse the x and y coordinates. Use the symmetry around y = x Example; page #38, 16, 18

The Horizontal Line Test For Inverse Functions  A function f has an inverse that is a function, f –1, if there is no horizontal line that intersects the graph of the function f at more than one point.

Example Does f(x) = x 2 +3x-1 have an inverse function?

Example Solution: This graph does not pass the horizontal line test, so f(x) = x 2 +3x-1 does not have an inverse function.

Finding the Inverse of a Function The equation for the inverse of a function f can be found as follows: 1.Verify the function is one-to-one (HLT). 2.Replace f (x) by y in the equation for f (x). 3.Interchange x and y. 4.Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function. 5.If f has an inverse function, replace y in step 3 with f - 1 (x). We can verify our result by showing that f ( f -1 (x)) = x and f -1 ( f (x)) = x.

Find the inverse of f (x) = 6x + 3. Solution Step 1 Replace f (x) by y. y = 6x + 3 Step 2 Interchange x and y. x = 6y + 3 This is the inverse function. Step 3 Solve for y. x - 3 = 6y Subtract 3 from both sides. x - 3 = y 6 Divide both sides by 6. Step 4 Replace y by f -1 (x). x f -1 (x) = Rename the function f -1 (x). Example

Examples Page 118 #58, 76 Page 118 #58, 76 Groups: Pg 117: # 12, 28, 30, 39, 41, 62 Groups: Pg 117: # 12, 28, 30, 39, 41, 62 Homework: Pg 117: # 1 – 41 odd, 63, 67, 75 Homework: Pg 117: # 1 – 41 odd, 63, 67, 75