( ) a (graphical) transformation of y = f ( x )? x 0,2π. f ( 1 b) = a if and only if f ( a ) = b. f 1 1 f

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1 Warm-Up: Solve sinx = 2 for x 0,2π 5 (a) graphically (approximate to three decimal places) y (b) algebraically BY HAND EXACTLY (do NOT approximate except to verify your solutions) x x 0,2π, xscl = π 6,y,, yscl = 2 Definition of an Inverse Function: If f is a one-to-one function with domain D and range R, then the inverse function of f, denoted f, is the function with domain R and range D defined by f ( b) = a if and only if f ( a ) = b. NOTE: f f the inverse IS NOT the reciprocal! f is the inverse or undoing function of f, whereas f is the reciprocal function of f. Answer the following questions in the space provided.. In what way is y = f x ( ) a (graphical) transformation of y = f ( x )? 2. What is the purpose of an inverse function? 3. Define: a) function c) vertical line test b) one-to-one function d) horizontal line test Raelene Dufresne 203 of 8

2 4. Solve for x, for x : a) x 3 = 8 b) x 2 = 6 c) 2 x = 8 d) log x = 0. e) sinx = 2 5. Sketch the following functions on the grids below and (i) identify which are one-to-one and which are not. (ii) sketch the inverse function for the one-to-one functions. (iii) restrict the domains of the non-one-to-one functions so that they are one-to-one, and state these restricted domains. Then sketch the inverse of these restricted domain functions. a) f ( x ) = x 3 b) f ( x ) = x 2 c) f ( x ) = 2 x f ( x ) = log x e) f ( x ) = sinx f) f ( x ) = cos x Raelene Dufresne of 8

3 6. The quantity that you take a function of is called the argument of the function. For example, the argument of f x ( ) is x. a) What kind of quantity is the argument of exponential functions? b) What kind of quantity results from an exponential function? c) Which function is the inverse function of the exponential function base b? What kind of quantity is its argument and what kind of quantity results from this function? d) What kind of quantity is the argument of the sine function? Of any trig function? e) What quantity results from the sine function? Of any trig function? f) What kind of quantity is the argument of the inverse sine function? g) What kind of quantity results from the inverse trig function? h) Rewrite 2 x = 5 in its inverse logarithmic statement: Let trig represent any of the six trigonometric functions (ratios). function and inverse function notation can be expressed as follows: Then the trig a) Trig function notation: trig( angle)=ratio trig θ ( ) = a b a b) Inverse trig function notation: trig ( - ratio)=angle trig - = θ b Example : Rewrite sin30 o = 2 as an inverse trig statement. Example 2: Evaluate the following (express any angles in radian, not degree measure). a) sin π b) sin 5π c) sin d) sin π e) sin 7π f) sin π g) sin Raelene Dufresne of 8

4 Exercises:. Who am I? Write the equation for each function on its graph. If the function is not one-to-one, state its restricted domain. 2. Evaluate the following limits: lim sin x x lim cos x x 0 lim tan x x lim cot x x lim tan x x lim cot x x Raelene Dufresne of 8

5 3. Evaluate the following: ( ) = sin30 o = sin50 o = sin a) sin 20 o = 2 b) sin 30 o ( ) = sin20 o = sin330 o = sin 2 = 4. Sketch the graph of each function and use the graph to evaluate the given value. 2 a) f ( x ) = x and 9 b) g ( x ) = sin x and sin 2 5. Fill in the blanks: a) The value of sin x is the between and inclusive/exclusive (circle the correct word) such that the of over is equal to. b) The value of cos x is the between and inclusive/exclusive (circle the correct word) such that the of over is equal to. 6. Evaluate exactly. a) cos 3 2 b) sin 2 2 c) sin d) tan tan 7π 2 7. Evaluate exactly. 2 3 a) sin sin b) sin sin 3 2 π 7 c) cos sec d) tan sec 2 8 Raelene Dufresne of 8

6 8. Write an algebraic expression (without using trig, trig -, exp or log functions) in x terms of x for sin cos. Determine a restriction on the value of x. x + 2 ( ) Evaluate: a) tan cos b) sin csc Evaluate exactly without a calculator. Show any work necessary. ( ) 5 a) sin cos b) sin sin ( ) c) cos cos 4.8. Evaluate exactly without a calculator: a) sin sin 5π b) cos cos 5π c) tan tan 5π d) sin cos π 3 Raelene Dufresne of 8

7 The Calculus of Inverse Trig Functions Given that you know the derivative of a function f, how can you find the derivative of its inverse? Well, in 4.2 we saw that derivatives of inverse functions are reciprocals, and that the input of one function is the output of its inverse (and vice versa). So to show that d dx f Let y = f ( x ). Then f y ( ) = x. ( x ) = f y ( ), start off with a substitution to eliminate f : ( ) = x Differentiate with respect to x: d dx f y Therefore: f ( y ) y = y = (Note: reciprocal relationship bw derivatives.) f ( y ) And as is usual with substitution, we back-substitute y = f ( x ) to leave the expression in terms of the original variable: y = 2. Show the following: a) ( ) = d dx sin x. f f ( x ) ( ) = x 2 c) d dx tan x e) d ( + x 2 dx sec x ) = x x 2 b) d ( dx cos x ) = ( ) = x 2 d) d dx cot x ( ) = f) d + x 2 dx csc x x x 2 Raelene Dufresne of 8

8 3. Determine the derivatives of the following functions: a) f x ( ) = sin ( sin x ) c) f ( x ) = cot ( secx ) for 0 < x < π 2 b) f x ( ) = tan x d) f ( x ) = csc ( e 2x + x ) 4. Determine the vector equation of the tangent line to the curve r ( t ) = sin ( 2t ) î + ln ( cos t ) ĵ when t = 0. Raelene Dufresne of 8