H.Melikyan/12001 Inverse Trigonometric Functions

H.Melikyan/12002 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.

H.Melikyan/12003 Text Example Show that each function is the inverse of the other: f (x) = 5x and g(x) = x/5. 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) = 5x f (g(x)) = 5g(x) = 5(x/5) = x. Next, we find g(f (x)). g(x) = 5/x g(f (x)) = f (x)/5 = 5x/5 = x. Notice how f -1 undoes the change produced by f.

H.Melikyan/12004 Finding the Inverse of a Function The equation for the inverse of a function f can be found as follows: 1. Replace f (x) by y in the equation for f (x). 2. Interchange x and y. 3. 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. 4. 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.

H.Melikyan/12005 Find the inverse of f (x) = 7x – 5. Solution Step 1 Replace f (x) by y. y = 7x – 5 Step 2 Interchange x and y. x = 7y – 5 This is the inverse function. Step 3 Solve for y. x + 5 = 7y Add 5 to both sides. x + 5 = y 7 Divide both sides by 7. Step 4 Replace y by f -1 (x). x f -1 (x) = Rename the function f -1 (x). Text Example

H.Melikyan/12006 The Horizontal Line Test For Inverse Functions v 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.

H.Melikyan/12007 The inverse sine function, denoted by sin -1, is the inverse of the restricted sine function y = sin x, -  /2 < x <  / 2. Thus, y = sin -1 x means sin y = x, where -  /2 < y <  /2 and –1 < x < 1. We read y = sin -1 x as “ y equals the inverse sine at x.” y 1  /2 /2 x -  / 2 y = sin x -  / 2 < x <  / 2 Domain: [ -  / 2,  / 2] Range: [ -1, 1 ] The Inverse Sine Function

H.Melikyan/12008 The Inverse Sine Function

H.Melikyan/12009 Finding Exact Values of sin -1 x

H.Melikyan/ Example v Find the exact value of sin -1 (1/2)

H.Melikyan/ The Inverse Cosine Function The inverse cosine function, denoted by cos -1, is the inverse of the restricted cosine function y = cos x, 0 < x < . Thus, y = cos -1 x means cos y = x, where 0 < y <  and –1 < x < 1.

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H.Melikyan/ Find the exact value of cos -1 (-  3 /2) Solution Step 1 Let  = cos -1 x. Thus  = cos -1 (-  3 /2) We must find the angle , 0 <  < , whose cosine equals -  3 /2 Step 2 Rewrite  = cos -1 x as cos  = x. We obtain cos  = (-  3 /2) Step 3 Use the exact values in the table to find the value of  in [0,  ] that satisfies cos  = x. The table on the previous slide shows that the only angle in the interval [0,  ] that satisfies cos  = (-  3 /2) is 5  /6. Thus,  = 5  /6 Text Example

H.Melikyan/ The Inverse Tangent Function The inverse tangent function, denoted by tan -1, is the inverse of the restricted tangent function y = tan x, -  /2 < x <  /2. Thus, y = tan -1 x means tan y = x, where -  /2 < y <  /2 and –  < x < .

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H.Melikyan/ Inverse Properties The Sine Function and Its Inverse sin (sin -1 x ) = x for every x in the interval [-1, 1]. sin -1 (sin x ) = x for every x in the interval [-  /2,  /2 ]. The Cosine Function and Its Inverse cos (cos -1 x ) = x for every x in the interval [-1, 1]. cos -1 (cos x ) = x for every x in the interval [0,  ]. The Tangent Function and Its Inverse tan (tan -1 x ) = x for every real number x tan -1 (tan x ) = x for every x in the interval (-  /2,  /2 ).