1. The Domain of f is the set of all allowable inputs (x values)
Functions
A function is an operation performed on an input (x) to produce an output (y = f(x) ).
The Domain of f is the set of all allowable inputs (x values)
The Range of f is the set of all outputs (y values)
Domain
Range f x
y =f(x)

2. To be well defined a function must
·       Have a value for each x in the domain
·       Have only one value for each x in the domain
e.g y = f(x) = √(x-1), x   is not well defined as if x < 1 we will be trying to square root a negative number.
y = f(x) = 1/(x-2), x   is not well defined as if x = 2 we will be trying to divide by zero.
This is not a function as some x values correspond to two y values.

3. Finding the Range of a function
Draw a graph of the function for its given Domain
The Range is the set of values on the y-axis for which a horizontal line drawn through that point would cut the graph.
The Function is f(x) = (x-2)2 +3 , x
y = (x-2)2 +3
The Range is
f(x) ≥ 3
Domain
y = (x-2)2 +3 2
Range 3
Link to Inverse Functions
Domain

4. The Function is f(x) = 3 – 2x , x
The Range is f(x) < 3

5. Composite Functions Finding gf(x) g f f(x) g(f(x)) = gf(x)f g
Note: gf(x) does not mean g(x) times f(x).
f(x) x
g(f(x))
= gf(x)
Note : When finding f(g(x))
Replace all the x’s in the rule for the f funcion with the expression for g(x) in a bracket.
e.g If f(x) = x2 –2x
then f(x-2) = (x-2)2 – 2(x-2)
gf(x) means “g of f of x” i.e g(f(x)) .
First we apply the f function.
Then the output of the f function becomes the input for the g function.
Notice that gf means f first and then g.
Example if f(x) = x + 3, x and g(x) = x2 , x then
gf(x) = g(f(x)) = g(x + 3) = (x+3)2 , x
fg(x) = f(g(x)) = f(x2) = x2 +3, x
g2(x) means g(g(x)) = g(x2) = (x2)2 = x4 , x
f2(x) means f(f(x)) = f(x+3) = (x+3) + 3 = x + 6 , x

6. Notice that fg and gf are not the same.
The Domain of gf is the same as the Domain of f since f is the first function to be applied.
The Domain of fg is the same as the Domain of g.
For gf to be properly defined the Range (output set) of f must fit inside the Domain (input set) of g.
For example if g(x) = √x , x ≥ 0 and f(x) = x – 2, x
Then gf would not be well defined as the output of f could be a negative number and this is not allowed as an input for g.
However fg is well defined, fg(x) = √x – 2, x ≥ 0.

7. Inverse Functions. Domain of f Range of f
 The inverse of a function f is denoted by f-1 .
The inverse reverses the original function.
So if f(a) = b then f-1(b) = a 
Note: f-1(x) does not mean 1/f(x). b f a
f-1
Domain of f
Range of f
= Domain of f-1
= Range of f-1

8. One to one Functions
If a function is to have an inverse which is also a function then it must be one to one.
This means that a horizontal line will never cut the graph more than once.
i.e we cannot have f(a) = f(b) if a ≠ b,
Two different inputs (x values) are not allowed to give the same output (y value).
For instance f(-2) = f(2) = 4
y = f(x) = x2 with domain x is not one to one.
So the inverse of 4 would have two possibilities : -2 or 2.
This means that the inverse is not a function.
We say that the inverse function of f does not exist.
If the Domain is restricted to x ≥ 0
Then the function would be one to one and its inverse would be
f-1(x) = √x , x ≥ 0

9. Finding the Rule and Domain of an inverse function
Swap over x and y
Make y the subject
Domain
The domain of the inverse = the Range of the original.
So draw a graph of y = f(x) and use it to find the Range
Drawing the graph of the Inverse
The graph of y = f-1(x) is the reflection in y = x of the graph of y = f(x).

10. Find the inverse of the function y = f(x) = (x-2)2 + 3 , x ≥ 2
Example:
Find the inverse of the function y = f(x) = (x-2)2 + 3 , x ≥ 2
Sketch the graphs of y = f(x) and y = f-1(x) on the same axes showing the relationship between them.
Domain
This is the function we considered earlier except that its domain has been restricted to x ≥ 2 in order to make it one-to-one.
We know that the Range of f is y ≥ 3 and so the domain of f-1 will be x ≥ 3.
Rule
Swap x and y to get x = (y-2)2 + 3
Now make y the subject
x – 3 = (y-2)2
√(x –3) = y-2
y = 2 + √(x –3)
So Final Answer is:
f-1(x) = 2 + √(x –3) , x ≥ 3
Graphs
Reflect in y = x to get the graph of the inverse function.
Note: we could also have
-√(x –3) = y-2
and y = 2 - √(x –3)
But this would not fit our function as y must be greater than 2 (see graph)
Note: Remember with inverse functions everything swaps over.
Input and output (x and y) swap over
Domain and Range swap over
Reflecting in y = x swaps over the coordinates of a point so (a,b) on one graph becomes (b,a) on the other.