Inverse functions

1
Inverse functions

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

2
What is it?

• INVERSE FUNCTION reversing a function,
  undoing.
• f -1 notates an inverse function. (not 1/f)

3
WHAT??

• Find the inverse of f(x)4x-2

4
-2
x
4x-2
(x2)/4
x
/4
2
So f -1 (x)(x2)/4
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 yx

5
y ex, y x and y ln x
y ex
y x
y ln (x)
yln(x) is a reflection of y ex in the line y
x
yln(x) and y ex are inverse functions
6
A neat little trick

• As always in maths, there is a trick to this
• Write function as a rule in terms of y and x.
• Swap x and y
• Rearrange to get in terms of y.

7
A neat little trick
Find the inverse of
8
Inverse functions with ex
e.g. f(x) ex -2
y ex -2
y 2 ex
ln(y 2) ln ex
ln(y 2) x
The inverse of f(x) is
f-1(x) ln(x 2)
Domain is x gt -2
9
Inverse functions with ex
e.g. f(x) e2x-1 6
y e2x-1 6
The inverse of f(x) is
y - 6 e2x-1
f-1(x) ½(ln(x-6) 1)
ln(y - 6) ln e2x-1
Domain ?
ln(y - 6) 2x - 1
Cannot have ln of numbers less than 0
ln(y - 6) 1 2x
Domain is x gt 6
½(ln(y - 6) 1) x
10
Inverse functions with ln x
e.g. f(x) ln(2x) 6
y ln(2x) 6
y - 6 ln (2x)
ey-6 eln 2x
The inverse of f(x) is
ey-6 2x
f-1(x) ½ ex-6
Domain ?
½ ey-6 x
11

Exercises

• page 84
• Exercise D
• Q1,2 a, c, e
• Q7
• Page 87
• Test Yourself
• Q7

12
Test yourself Q7 AQA 2001
y
f(x) 2e-x
x
a) Describe the set of transformations to get
from y ex
c) Find f-1(x)
b) Sketch y 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.
13
Test yourself Q7 AQA 2001
y
f(x) 2e-x
x
a) Describe the set of transformations to get
from y ex
reflection in the y axis
Stretch by factor 2 in the y-direction
14
Test yourself Q7 AQA 2001
y
f(x) 2e-x
x
b) Sketch y f-1(x)
c) Find f-1(x)
y 2e-x
d) What is the domain for f-1(x)
y/2 e-x
Domain is x gt 0
ln (y/2) ln e-x
ln is always gt 0
ln (y/2) -x
f-1(x) -ln(x/2)
-ln (y/2) x
15
Test yourself Q7 AQA 2001
mg
f(x) 2e-x
t
f(t) 2e-t
0.5
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.
2e-t 0.5
e-t 0.25
ln e-t ln 1/4
-t ln 4-1
-t - ln 4
t ln 4
1.39 hrs