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Inverse Trigonometric Functions H.Melikyan/1200 1 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/1200 2 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/1200 3 Finding the Inverse of a Function The equation for the inverse of a function f can be found as follows: 1. 2. 3. 4. Replace f (x) by y in the equation for f (x). Interchange x and y. 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. 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/1200 4 Text Example Find the inverse of f (x) = 7x – 5. Solution Step 1 Replace f (x) by y. y = 7x – 5 Step 2 Step 3 Step 4 Interchange x and y. x = 7y – 5 This is the inverse function. x + 5 = 7y x+5=y 7 Add 5 to both sides. Solve for y. Replace y by f -1(x). f -1(x) = H.Melikyan/1200 x+5 7 Divide both sides by 7. Rename the function f -1(x). 5 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. H.Melikyan/1200 6 The Inverse Sine Function 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 y = sin x - /2 < x < /2 1 - /2 x /2 -1 H.Melikyan/1200 Domain: [- /2, /2] Range: [-1, 1] 7 The Inverse Sine Function H.Melikyan/1200 8 Finding Exact Values of sin-1x H.Melikyan/1200 9 Example Find the exact value of sin-1(1/2) sin 1 1 2 1 sin 2 1 sin 6 2 H.Melikyan/1200 6 10 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. H.Melikyan/1200 11 H.Melikyan/1200 12 H.Melikyan/1200 13 Text Example 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 H.Melikyan/1200 14 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 < . H.Melikyan/1200 15 H.Melikyan/1200 16 H.Melikyan/1200 17 H.Melikyan/1200 18 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). H.Melikyan/1200 19