A function is a rule which indicates an operation to perform. e.g:

if f(x) = x² + 3

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. 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 y = ax² + bx + c . They can be 'U' shaped, when a is greater than zero, or 'n' shaped, when a is less than zero.

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 funtion does the opposite of the function. To find the inverse of a function, follow the following procedures: let y = f(x). Swap all y's and x's . Rearrange to give y = . This is the inverse function.

Example:

f(x) = 3x - 7

y = 3x - 7 (let f(x) = y)

x = 3y - 7 (swap x's and y's)

y = x + 7

3

Combining Functions

If f(x) = 3x + 1 and g(x) = x² + 2

therefore f(x) + g(x) = 7

so 3x + 1 + x² + 2 = 7

so (x - 1)(x + 4) = 0

so x = 1 or -4