## Functions

This section looks at functions within the wider topic of Algebra.

A **function** may be thought of as a rule which takes each member **x** of a set and assigns, or maps it to the same value **y** known at its image.

**x → Function → y**

A letter such as *f, g or h *is often used to stand for a function. The Function which squares a number and adds on a 3, can be written as** f(x) = x^{2}+ 5**. The same notion may also be used to show how a function affects particular values.

*Example*

*f*(4) = 4^{2 }+ 5 =21, *f*(-10) = (-10)^{2} +5 = 105 or alternatively *f*: **x → x ^{2 }+ 5**.

The phrase "y is a function of x" means that the value of y depends upon the value of x, so:

- y can be written in terms of x (e.g. y = 3x ).
- If f(x) = 3x, and y is a function of x (i.e. y = f(x) ), then the value of y when x is 4 is f(4), which is found by replacing x"s by 4"s .

**Example**

If f(x) = 3x + 4, find f(5) and f(x + 1).

f(5) = 3(5) + 4 = __19__

f(x + 1) = 3(x + 1) + 4 = __3x + 7__

**Domain and Range**

The **domain **of a function is the set of values which you are allowed to put into the function (so all of the values that x can take). The **range **of the function is the set of all values that the function can take, in other words all of the possible values of y when y = f(x). So if y = x^{2}, we can choose the domain to be all of the real numbers. The range is all of the real numbers greater than (or equal to) zero, since if y = x^{2}, y cannot be negative.

**One-to-One**

We say that a function is **one-to-one** if, for every point y in the range of the function, there is only one value of x such that y = f(x). f(x) = x^{2} is not one to one because, for example, there are two values of x such that f(x) = 4 (namely –2 and 2). On a graph, a function is one to one if any horizontal line cuts the graph only once.

**Composing Functions**

fg means carry out function g, then function f. Sometimes, fg is written as fog

**Example**

If f(x) = x^{2} and g(x) = x – 1 then

gf(x) = g(x^{2}) = x^{2} – 1

fg(x) = f(x – 1) = (x – 1)^{2}

As you can see, fg does not necessarily equal gf

**The Inverse of a Function**

The inverse of a function is the function which reverses the effect of the original function. For example the inverse of y = 2x is y = ½ x .

To find the inverse of a function, swap the x"s and y"s and make y the subject of the formula.

**Example**

Find the inverse of f(x) = 2x + 1

Let y = f(x), therefore y = 2x + 1

swap the x"s and y"s:

x = 2y + 1

Make y the subject of the formula:

2y = x - 1, so y = ½(x - 1)

Therefore f ^{-1}(x) = __½(x - 1)__

f^{-1}(x) is the standard notation for the inverse of f(x). The inverse is said to exist if and only there is a function f^{-1} with ff^{-1}(x) = f^{-1}f(x) = x

Note that the graph of f^{-1} will be the reflection of f in the line y = x.

**Graphs**

Functions can be graphed. A function is **continuous **if its graph has no breaks in it. An example of a discontinuous graph is y = 1/x, since the graph cannot be drawn without taking your pencil off the paper:

A function is **periodic** if its graph repeats itself at regular intervals, this interval being known as the period.

A function is **even** if it is unchanged when x is replaced by -x . The graph of such a function will be symmetrical in the y-axis. Even functions which are polynomials have even degrees (e.g. y = x²).

A function is **odd** if the sign of the function is changed when x is replaced by -x . The graph of the function will have rotational symmetry about the origin (e.g. y = x³).

**The Modulus Function**

The modulus of a number is the magnitude of that number. For example, the modulus of -1 ( |-1| ) is 1. The modulus of x, |x|, is x for values of x which are positive and -x for values of x which are negative. So the graph of y = |x| is y = x for all positive values of x and y = -x for all negative values of x:

**Transforming Graphs**

If y = f(x), the graph of y = f(x) + c (where c is a constant) will be the graph of y = f(x) shifted c units upwards (in the direction of the y-axis).

If y = f(x), the graph of y = f(x + c) will be the graph of y = f(x) shifted c units to the left.

If y = f(x), the graph of y = f(x – c) will be the graph of y = f(x) shifted c units to the right.

If y = f(x), the graph of y = af(x) is a stretch of the graph of y = f(x), scale factor (1/a), parallel to the x-axis. [Scale factor 1/a means that the "stretch" actually causes the graph to be squashed if a is a number greater than 1]

**Example**

The graph of y = |x - 1| would be the same as the above graph, but shifted one unit to the right (so the point of the V will hit the x-axis at 1 rather than 0).