Harder one:

A function is defined by : f(x) = (x+3)(x-5)

Find the inverse and state the domain and range of f(x) and f-1(x)

A function is defined by : f(x) = (x-16)(x-4)

Find the inverse and state the domain and range of f(x) and f-1(x)

Homework:

Functions on moodle with mark scheme

Inverse functions

Recap of inverse of a function.Inverse functions with ex and ln

xHarder inverse functions such

as quadratics and algebraic fractions

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.4. Write as f-1 (x) =

A neat little trick…

32 xxf

32 xy

32 yx

62 yx

yx 26

2

6x

y

Find the inverse of

2

61 x

xf

Inverse functions

Inverse functions only exist for one-one functions.

Things to note..

The domain of f-1 is the range of f and the range of f-1 is the domain of f.

The graph of an inverse function can be found by reflecting a function in the line y=x

Have a go:

Worksheet C

Questions 1-7

ppt: 10 questions

-5

-4

-3

-2

-1

0

1

2

3

4

5

-3 -2 -1 0 1 2 3

y=ln(x) is a reflection of y = ex in the line y = x

y = ex

y = x

y = ln (x)

y = ex, y = x and y = ln x

y=ln(x) and y = ex are inverse functions

Inverse functions with ex

e.g. f(x) = ex -2 x = ey -2

x +2 = ey

ln(x +2) = ln ey

ln(x +2) = y

The inverse of f(x) is …

f-1(x) = ln(x +2)

Domain is x > -2 -5

-4

-3

-2

-1

0

1

2

3

4

5

-3 -2 -1 0 1 2 3

y = ex - 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 ex

e.g. f(x) = e2x-1 + 6 x = e2y-1 + 6

x - 6 = e2y-1

ln(x - 6) = ln e2y-1

ln(x - 6) = 2y - 1

The inverse of f(x) is …

f-1(x) = ½(ln(x-6) + 1)

Domain ?

ln(x - 6) +1 = 2y

½(ln(x - 6) +1) = y Domain is x > 6

Cannot have ln of numbers less than 0

Inverse functions with ln x

e.g. f(x) = ln(2x) + 6 x = ln(2y) + 6

x - 6 = ln (2y)

ex-6 = eln 2y

ex-6 = 2y

The inverse of f(x) is …

f-1(x) = ½ ex-6

Domain ? ½ ex-6 = y

Have a Go

Domino Trailor

Worksheet C

Questions 8,9,11

Extension: Exam Questionsppt: 10 questions

Plenary

Plenary

f(x) = x . x+1

Show that f-1(x) = 1 - 1 1-x

Extension: