A function indicates the relationship between variables. If *y* is a function of *x*, the value of *y* depends on the value of *x* and for each value of *x* there is only one value of *y*.

In other words, think of a function as a rule of correspondence assigning a single value of *y* to every value of *x*.

The equation *y* = *x* + 3 can also be thought of as a function, where the value of *y* depends on the value of *x*. Each value of *x* yields only one value of *y*.

We write the function as: f(*x*) = *x* + 3

In function notation, ** y** is replaced by

f (*x*) is used to denote a function of *x* and it is read as ‘ f of *x*’.

f(*x*) is also called the function value of * x* under f.

The letter* x* stands for the variable of f.

When the variable *x* is given a value, f(*x*) will then have a value. For example, f(1) is the function value under f when *x* takes on the value 1.

* Example: *

Given f(*x*) = *x*^{2} + 2*x*, find the value of f(3)

* Solution:*

f(3) = (3)^{2} + 2(3) = 9 + 6 = 15

A composite function is a composition of 2 or more functions into a single function.

Let us look at a composite function pictorially. Suppose f and g are 2 functions. The diagram below describes what the composite function gf is.

The 2 short arrows represent f and g respectively. First f maps value *x* to function value f(*x*). Then g maps value f(*x*) to function value g(f(*x*)). In this case, the function value f(*x*) becomes the input for the variable in g(*x*) for the second arrow. The resulting value under function g is g(f(*x*)).

The long arrow represents gf as a single function. This function maps *x* directly to the function value g(f(*x*)).

In other words, the long arrow produces the same result as the end result of the 2 short arrows in 2 “hops”. gf is called the composite function of f and g.

* Example: *

Given f (*x*) = 2*x* + 1 and g(*x*) = *x*^{2} – 2, find: gf (5)

* Solution: *

gf (5) = g( f(5))

= g(2(5) + 1)

= g(11)

= 11^{2} – 2

= 119

The following videos show more examples of functions and composite functions.