1
Functions and Graphs
You should be able to sketch these graphs:
y = x2
and
y = x3
y = x2
y = x3
Maths revision course by Miriam Hanks
2
Functions and graphs
You should be able to sketch these graphs:
y= 1
and
y = tan x
x
y = tan x
y = 1/x
Maths revision course by Miriam Hanks
3
Functions and graphs
You should be able to sketch these graphs:
y = sin x
and
y = cos x
y = sin x
y = cos x
Maths revision course by Miriam Hanks
4
Functions and graphs
You should be able to sketch these graphs:
y = ex
and
y = log x
y = log x
y = ex
Maths revision course by Miriam Hanks
5
Functions – Domain and Range



What are domain and range?
Domain is all possible x - values
Range is all possible y-values
eg What are the domain and range of y = ex ?
The domain is x ε R, and the range is f(x) > 0.
Maths revision course by Miriam Hanks
6
Functions – Domain and Range

The domain of a fraction is restricted:
Since we can NEVER divide by zero, the
denominator of a fraction cannot be zero, so
for example, y = (3x + 2) / (x – 1) is
restricted because
x cannot be equal to 1, ie the domain is
x=1
Maths revision course by Miriam Hanks
7
Inverse functions f
eg Find the inverse
function of f(x) = 2x + 1
Swap x and y:
x = 2y + 1
then rearrange to make y
the subject:
y = (x – 1) /2
–1
(x)
To find an inverse
function:

swap x and y, then
rearrange it to
make y the subject
So f –1(x) = (x – 1 )/2
Maths revision course by Miriam Hanks
8
Inverse functions f
eg A function f(x) = x2
has range f(x) > 0.
What is the domain of
the inverse function ?
(x)
To find the domain
and range of an
inverse function:

x>0
–1
swap the domain
and range of the
original function
Maths revision course by Miriam Hanks
9
Inverse functions f
–1
(x)
To sketch an
inverse function:
f(x)

f -1(x)
reflect the original
function in the
diagonal line y = x
Maths revision course by Miriam Hanks
10
Composite functions


f(x ) = x2 + 4x
g(x) = 2x + 3
To find fg(x), pick
up g and put it into
f in place of each x.
What is fg(x) ?
fg(x) = (2x + 3)2 + 4(2x +3)
= 4x2 + 12x + 9 + 8x + 12
= 4x2 + 20x + 21
Maths revision course by Miriam Hanks
11
Transformations of functions

f(x) + 5 moves UP 5 units
y = x2 + 5

f(x + 5) moves LEFT 5 units
y = (x +
5)2
y = x2
Maths revision course by Miriam Hanks
12
Transformations of functions

- f(x)reflects in the x- axis
y = e-x

f(-x) reflects in the y-axis
y = ex
y =-ex
Maths revision course by Miriam Hanks
13
Transformations of functions

2f(x) multiplies the y-coordinates by 2

f(2x) divides the x-coordinates by 2 (and so you
have twice as many waves)
y = sin x
y = sin (2x)
y =2 sin (x)
Maths revision course by Miriam Hanks