Function Notation Is Another Way of Writing Equations

Functions

Function notation is another way of writing equations.

For example: instead of writing , we could write

(See lesson 2 for more information about function notation).

is the INPUT and is called the DOMAIN.

is the OUTPUT and is called the RANGE.

A function should only have one -value for every -value.

is a function because when you put a value in there is only one possible answer.

isnot a function as each value you put in has 2 possible answers (e.g. ).

Evaluating functions with numbers:

We looked at evaluating functions in lesson 2, but here is a recap: If you are asked to evaluate a function you need to put in the -value (domain) and find the resulting -value (range).

Examples:

Given the function:

Find

This just means wherever you see an replace it with an8.

Given the function:

Find

Now wherever you see an replace it with a 3.

Evaluating functions with Algebra:

Whatever you are asked to find the function of just replace the with it, no matter how complicated.

Examples:

Given the function:

Find

Wherever you see an replace it with.

Replace with :

Tidy up:

Given the function:

Find

Wherever you see an replace it with .

Replace with :

Multiply all brackets out:

Tidy up:

Given the function:

Find

Wherever you see an replace it with .

Replace with :

Multiply out bracket:

Tidy up:

Composite Functions:

You may be asked to combine 2 functions. This can be written asor. To work it out you apply (the inside function) first thenevaluate. If you were asked to do you would apply (the inside function) first thenevaluate.

Examples:

Given the functions:and

Find:

Find :

Find :11

Find :

Find :35

this is the same as doing

Find :

Find :5

Find :

Inverse Functions:

If you have a function then the notation for the inverse function is .You find an inverse function by “undoing” the original function. So if the original ended up with “add 3” then the inverse would start with “subtract 3” etc.

This is the method to find the inverse:

Replace the function notation with
Rearrange the function to make the subject
Replace with and vice versa.
Put the function notation back in.

(As we are undoing the original function no matter what is .)

Work through the following examples for more guidance on finding inverse functions.

Examples:

Given the function: , find

Replace the function notation with :

Rearrange the function to make the subject:

(See Lesson 3 for help with rearranging)

Replace with and with :

Put the function notation back in:

Given the function: , find

Replace the function notation with :

Rearrange the function to make the subject:

Replace with and with :

Put the function notation back in:

Asymptotes:

Some functions have certain values of which cannot be used.

For example:

If you were to plot this function on a graph then there would be a line at which the graph cannot cross, this line is known as an asymptote.

Finding the Domain and the Range:

The idea of asymptotes can be used to help to find the domain of a function.

Domain

The domain is all the values that can go into the function.

Probably the easiest way to find the domain is to look for any values that can’t go in.

In the example above the domain would be: all real numbers except 2 (i.e. )

The main things to look for are:can’t divide by 0

can’t square root a negative number

Examples:

Find the domain of the function

(thinkof any values that cannot be put into the function)

Domain:( can be anything except -5)

Find the domain of the function

Domain:( has to be 3 or above so that the number in the square root sign isn’t negative)

Range

The range is all the values that can come out of the function.

To find the range just use common sense and think if there is anything that can’t come out of the function. Also a function should only have one possible answer so you need to limit the range to validate the function.

Examples:

Find the range of the function

(thinkof any values that cannot come out of the function)

Range:(not possible to divide 3 and end up with 0)

Find the range of the function

Range:(By specifying that the answer has to be greater than 0 we have validated the function as there is now only 1 possible answer for each value)