A function and its inverse function? The function f(x)= x+1 is one-to-one for x? > ? 1 and has an inverse on that interval. a.? Graph ?f? for ?x? > ?1. ?1 b.? Find the inverse function f corresponding to the function graphed in part (a). Graph f ?1 on the same set of axes as in part (a). ?1 c.? Evaluate the derivative of f ? t the point . ?1 d.? Sketch the tangent lines on the graphs of ?f? and f at and , respectively.

Solution Step 1: Given f(x)= x x+1 Step 2: Graph of f for x>-1 is given below Step 3: (b)let y= f(x)= x+1 We can write it as (x + 1)y = x xy + y = x x(y 1) = y x = y (y1) x = y 1y Step 4: Write the function in the form y=f (x)(that is reversing the role of x and y in (1)) 1 y=f (x)= x 1x 1 Now the derivative of f (x)is given by d 1 d x dx f (x) = dx ( 1x ) (1x)dx(x)(xdx(1x) = 2 (1x) = (1x)(1)(x)(1) (1x) = 1 2 (1x) Step 5: The graph of f (x)=1 x is below 1x Step 6: 1 (c) from (b) the derivative of f (x) is d 1 1 1 dx f (x)=(f (x)) = 2 (1x) Step 7 (f (x)) at ( ,1)is 2 (f (x)) = 11 2 (12) 1 = ( ) 4 =4 Step 8: 1 1 1 (d) we have to sketch the tangent lines on the graph f and f at (1, 2 and ( ,1)2 The equation of tangent line at (x ,y ) w0th 0lope m is given by (y-y 0=m(x-x ) 0 1 1 1 Therefore to sketch the tangent lines on the graph f and f at (1, 2 and ( ,12 we need to find 1 1 the slopes at (1, ) 2nd ( ,1) 2 x f(x)= x+1 d x f’(x)= dx ( 1+x ) (1+x)d (x)(x)d(1+x) = dx dx (1+x)2 (1+x)(1)(x)(1) = (1+x) 1 =(1+x)2