Function f. Function f -1

Transcription

1 Page 1 REVIEW (1.7) What is an inverse function? Do all functions have inverses? An inverse function, f -1, is a kind of undoing function. If the initial function, f, takes the element a to the element b, then the inverse function takes the element b back to the element a. The domain of f = the range of f -1. The range of f = the domain of f -1. The graph of f and the graph of f -1 are symmetric with respect to the line y=x. (For example, if (a,b) is on the graph of f(x), then (b,a) is on the graph of f -1 (x). The composite of one function with its inverse becomes the identity function. That is, if you input x in the f function machine, you output f(x). If you input f(x) in the f -1 function machine, you output x. f -1 (f(x)) = x and also f - (f -1 (x)) = x Input x Function f Input f(x) Function f -1 Output f(x) * Note : f 1 Output x ( x) 1 f ( x)

2 Only one-to-one functions have inverses. Not all functions have inverses because all functions are not one-to-one functions. Definition of a one-to-one function: A function is a one-to-one if no two different elements in the domain have the same element in the range. The definition of a one-to-one function can be written algebraically as follows: A function f(x) is one-to-one if x 1 is not equal to x (x 1 and x any elements of the domain) then f(x 1 ) is not equal to f(x ). In other words, for any two ordered pairs (x 1,y 1 ) and (x, y ), where y 1 = f(x 1 ) and y = f(x ), Then if x 1 x, then y 1 y. Similarly, if f(x 1 )= f(x ), then it must be that x 1 = x. Just as we had a vertical line test to test if a graph represents a function, there is a horizontal line test to test if a function is 1-to-1. Horizontal Line Test Theorem If every horizontal line intersects the graph of a function f in at most one point, then f is 1-to-1. Below is the graph of y=x -4 (-4,1) However, for each x there is only one possible y, so y=x - 4 is a function For y=1, there are two possible x s. x=-4, and x=4. (4,1) A function is 1-to-1 over a certain interval only if it is constantly decreasing or constantly increasing over that interval. y=x -4 is 1-to-1 over the intervals (-,0) and (0, ) Page Does not pass Horizontal Line Test Therefore, this function is not 1-to-1. What would the inverse fuction of y = x -4 be? Solve for x. y + 4 = x x = ± y + 4 Which one do we choose? We need to have a specific value

3 Example 6 on p. 83 Finding the inverse function Find the inverse of f(x) = x + 3. Step 1: let y = f(x) and solve for y. y = x + 3 y 3 = x x = y/ 3/ = ½ (y-3) Page 3 Step : Interchange x and y and set y = f -1 (x) y = ½ (x-3) f -1 (x) = ½ (x-3) = inverse function Step 3: Check the result by showing that f -1 (f(x)) = x and also f - (f -1 (x)) = x Plug in f(x) = x + 3 f -1 (f(x)) = f -1 (x+3) = ½ ((x+3) 3) = ½ (x + 3 3) = ½ (x) = x f - (f -1 (x)) = f(½ (x-3)) = (½ (x-3)) + 3 = 1(x-3) + 3 = x = x y=x + 3 y=1/(x-3) y=x (1,5) y = x f(x) = x + 3 f -1 (x) = ½ (x-3) (5,1)

4 3.1 The Inverse Sine, Cosine, and Tangent Functions Let s look at f(x) = sin x The domain is all real numbers (which will represent angles). The range is the set of real numbers where -1 sin x 1. However, in order for the sine function to have an inverse function, it has to be 1-to-1. y = sin x Page (π/, 1) (5π/, 1) (0,0) (π, 0) (π, 0) Does not Pass Horizontal Line Test (-π/, -1) {-π/ x π/} passes Horizontal Line Test(3π/, -1) π π If we restrict the domain of y = sin x to the interval, then it will have an inverse function. The inverse sine function is denoted f -1 (x) = sin -1 x. The input of the inverse function is a real number between -1 and 1, and the output of inverse sine is a real number that is an angle in radians between π/ and π/. The domain is the set of real numbers {x -1 x 1}. π 1 π The range is the set of real numbers such that sin x By the property of inverse functions, for any x in the domain of sin x, sin -1 (sin (x)) = x and sin(sin -1 (x)) = x Notice that the input of the sine function is an angle, and the output of the inverse sine function is an angle (in radians). The inverse sine function also called the arcsin function. NOTE: sin -1 x 1 / sin x

5 Page 5 Example 3 on p.0 Find the exact value of sin -1 ( -½ ) What this is asking is, for what angle, θ, (where -π/ θ π/) does sin θ = ½? Let s look at the unit circle, x + y = 1. Remember each point on the unit circle is (cos θ, sin θ). So find a point on the unit circle with y-coordinate = -½ sin θ = - ½ when θ= -π/6 Therefore sin -1 (- ½ ) = -π/6 You try Exercise #7 on p.08 What is sin -1?

6 The Inverse Cosine Function In order for the cosine function to be one-to-one we must restrict its domain to {x 0 x π}. Page 6 y = cos x , π/6, π/6, {0 x π} passes Horizontal Line Test π, π/6, π/6, π/3, π/3, π/3, π/3, π/, π/, π/, π/, π/3, π/3, π/3, π/3, π/6, π/6, π/6, π, π/6, π, The inverse cosine function is denoted f -1 (x) = cos -1 x. The output of the inverse cosine function is a real number that is an angle in radians. The domain (input) is the set of real numbers {x -1 x 1}. The range (output) is the set of real numbers such that 1 0 cos x π { } By the property of inverse functions, for any x in the domain of cos x, cos -1 (cos (x)) = x and cos(cos -1 (x)) = x The inverse cosine function is also called the arccos function. Example 6 Find the exact value of cos 1 Look on the unit circle of an angle between 0 and π that gives a cosine value of

7 The Inverse Tangent Function In order for the tangent function to be one-to-one we must restrict π π its domain to < x < π. Notice that x cannot equal ± because tan x is undefined at those x-values. Page 7 {-π/ <x < π/} passes Horizontal Line Test 1 (π/4, 1) (5π/4, 1) (9π/4, 1) -π/ 0 π/ π 3π/ π 5π/ (-π/4, -1) -1 (3π/4, -1) (7π/4, -1) - Notice that the range of the tangent function is all real numbers. The graph extends to - when close to π/ and to + when close to π/. Therefore, for tan -1 x The domain is all real numbers. The range is the set of real numbers such that By the property of inverse functions, for any x in the domain of tan x, tan -1 (tan (x)) = x and tan(tan -1 (x)) = x Example 8 on p.07 Find the exact value of Hint, remember tan x = tan ( 3 ) 1 sin cos x x π < tan x < 1 π Now you do Exercise #3 on p.08

8 Page 8 HOMEWORK p. 08 Concepts and Vocab: #4,9 Exercises #3,5,11,13,15,17,5,6,9,35 Extra Credit (+ pts) #45 (all work must be shown)