Inverse functions  Recap of inverse of a function.  Inverse functions with e x and ln x

What is it?  INVERSE FUNCTION – reversing a function, “undoing”.  f -1 notates an inverse function. (not 1/f)

WHAT?? Find the inverse of f(x)=4x-2 x *4 4x-2 -2 (x+2)/4 x/4+2 So f -1 (x)=(x+2)/4

Things to note..  The domain of f -1 is the range of f.  The graph of an inverse function can be found by reflecting a function in the line y=x

y=ln(x) is a reflection of y = e x in the line y = x y = e x y = x y = ln (x) y = e x, y = x and y = ln x y=ln(x) and y = e x are inverse functions

A neat little trick…  As always in maths, there is a trick to this… 1. Write function as a rule in terms of y and x. 2. Swap ‘x’ and ‘y’ 3. Rearrange to get in terms of y.

A neat little trick… Find the inverse of

Inverse functions with e x e.g. f(x) = e x -2 y = e x -2 y +2 = e x ln(y +2) = ln e x ln(y +2) = x The inverse of f(x) is … f -1 (x) = ln(x +2) Domain is x > -2 y = e x - 2 y = ln (x+2) “The graph of an inverse function can be found by reflecting a function in the line y=x”

Inverse functions with e x e.g. f(x) = e 2x y = e 2x y - 6 = e 2x-1 ln(y - 6) = ln e 2x-1 ln(y - 6) = 2x - 1 The inverse of f(x) is … f -1 (x) = ½(ln(x-6) + 1) Domain ? ln(y - 6) +1 = 2x ½(ln(y - 6) +1) = x Domain is x > 6 Cannot have ln of numbers less than 0

Inverse functions with ln x e.g. f(x) = ln(2x) + 6 y = ln(2x) + 6 y - 6 = ln (2x) e y-6 = e ln 2x e y-6 = 2x The inverse of f(x) is … f -1 (x) = ½ e x-6 Domain ? ½ e y-6 = x

f(x) = 2e -x x y a) Describe the set of transformations to get from y = e x b) Sketch y = f -1 (x) c) Find f -1 (x) d) What is the domain for f -1 (x) e) A time t hours after an injection, a hospital patient has f(t) milligrams per litre of a certain drug in his blood. Find the time after the injection at which the patient has 0.5 milligrams per litre of the drug in his blood.

f(x) = 2e -x x y a) Describe the set of transformations to get from y = e x e x e -x reflection in the y axis e -x 2e -x Stretch by factor 2 in the y-direction

f(x) = 2e -x x y b) Sketch y = f -1 (x) “The graph of an inverse function can be found by reflecting a function in the line y=x” y = x c) Find f -1 (x) y = 2e -x y / 2 = e -x ln (y / 2 ) = ln e -x ln (y / 2 ) = -x -ln (y / 2 ) = x f -1 (x) = -ln(x/2) d) What is the domain for f -1 (x) Domain is x > 0 ln is always > 0

f(x) = 2e -x t mg e) A time t hours after an injection, a hospital patient has f(t) milligrams per litre of a certain drug in his blood. Find the time after the injection at which the patient has 0.5 milligrams per litre of the drug in his blood. f(t) = 2e -t = 0.5 2e -t = 0.5 e -t = 0.25 ln e -t = ln 1/4 -t = ln t = - ln 4 t = ln 4 = 1.39 hrs