Introduction
A function (or a 'map') is a rule which indicates an
operation to perform. You can think of a function as a box which you put numbers
into and get different ones out of. For example, a function might double any
number which you put into it.
Functions are usually written in the form f(x) = something.
The function which doubles any number you put into it is written f(x) = 2x. So
if you put 3 into the function, you get 6 out (2 times 3).
e.g. if f(x) = x² + 3
then f(2) = 2² + 3 = 7 (i.e.
replace x with 2)
Functions can be graphed. For
example, the graph of f(x) = 1/x is as follows:
This is the same graph as y = 1/x, although the y axis is
f(x) instead of y.
Types of graphs
The graph of y = k/x (f(x) = k/x) is known as a
hyperbola, where k is a constant (a fixed number). Asymptotes are
lines on a graph which the graph gets very close to, but never touches.
Therefore in the case of y = 1/x, the x and y axes are
asymptotes. Parabolas are graphs of the form y = ax² + bx + c (where
a, b and c are numbers). They can be 'U' shaped, when a is positive, or 'n'
shaped, when a is negative.
Graph Shifting
If you add 1 to f(x), this will shift the graph up 1 unit.
i.e. f(x) + n shifts the graph upwards by n units. f(x  1) will shift the
graph 1 unit to the right. i.e. f(x  n) shifts the graph n units to the
right. f(x + n) will shift the graph n units to the left.
Inverse Functions
The inverse function of y = 2x is y = ½x . The inverse of a
function does the opposite of the function. To find the inverse of a function,
follow the following procedure: let y = f(x). Swap all y's and x's . Rearrange
to give y = . This is the inverse function.
Example
Find the inverse of f(x), where f(x) = 3x  7 f(x) = 3x 
7 y = 3x  7 (let f(x) = y) x = 3y  7 (swap x's and y's) \ y = x + 7
3
So the inverse function is (x + 7)/3
Combining Functions
Let f(x) = 3x + 1 and g(x) = x² + 2 Suppose we are told
that f(x) + g(x) = 7 .
Then 3x + 1 + x² + 2 = 7
\x^{2} + 3x
 4 = 0 \ (x  1)(x + 4) =
0 \ x = 1 or –4
Copyright © Matthew Pinkney 2003
