Operations with Functions

add, subtract, multiply and divide

You can add, subtract, multiply and divide functions!

The result will be a new function.

Let us try doing those operations on f(x) and g(x):

add

Addition

You can add two functions:

(f+g)(x) = f(x) + g(x)

Note: I put the f+g inside () so you know they both work on x.

Example: f(x) = 2x+3 and g(x) = x2

(f+g)(x) = (2x+3) + (x2) = x2+2x+3

Sometimes you may need to combine like terms:

Example: v(x) = 5x+1, w(x) = 3x-2

(v+w)(x) = (5x+1) + (3x-2) = 8x-1

The only other thing to worry about is the Domain (the set of numbers that go into the function), but I will talk about that later!

 

Subtract

Subtraction

You can also subtract two functions:

(f-g)(x) = f(x) - g(x)

Example: f(x) = 2x+3 and g(x) = x2

(f-g)(x) = (2x+3) - (x2)

 

Multiply

Multiplication

You can multiply two functions:

(f·g)(x) = f(x) · g(x)

Example: f(x) = 2x+3 and g(x) = x2

(f·g)(x) = (2x+3)(x2) = 2x3 + 3x2

 

Divide

Division

And you can divide two functions:

(f/g)(x) = f(x) / g(x)

Example: f(x) = 2x+3 and g(x) = x2

(f/g)(x) = (2x+3)/x2

Function Composition

There is another special operation called Function Composition, read that page to find out more! (g º f)(x)

Domain

It has been very easy so far, but now you must consider the Domains of the functions.

doman and range

The domain is the set of all the values that go into a function.

The function must work for all values you give it, so it is up to you to make sure you get the domain correct!

 

Example: the domain for √x (the square root of x)

You cannot have the square root of a negative number (unless you use imaginary numbers, but we aren't doing that here), so we must exclude negative numbers:

The Domain of √x is all non-negative Real Numbers

On the Number Line it looks like:

zero onwards

Using set-builder notation it is written:

{ xmember ofreals | x ≥ 0}

Or using interval notation it is:

[0,+∞)

It is important to get the Domain right, or you will get bad results!

So how do you work out the new domain after doing an operation?

How to Work Out the New Domain

When you do operations on functions, you end up with the restrictions of both.

chicken dish

It is like cooking for friends:

  • one can't eat peanuts,
  • the other can't eat dairy food.

So what you cook can't have peanuts and also can't have dairy products.

Here is an example:

Example: f(x)=√x and g(x)=√(3-x)

The domain for f(x)=√x is from 0 onwards:

zero onwards

The domain for g(x)=√(3-x) is up to and including 3:

zero onwards

The new domain (after adding or whatever) is therefore from 0 to 3:

zero onwards

If you choose any other value, then one or the other part of the new function won't work.

In other words you want to find where the two domains intersect.

Note: we can put this whole idea into one line using Set Builder Notation:

Dom(f+g) = { xmember ofreals | xmember ofDom(f) and xmember ofDom(g) }

Which says "the domain of f plus g is the set of all Real Numbers that are in the domain of f AND in the domain of g"

The same rule applies when you add, subtract, multiply or divide, except divide has one extra rule.

An Extra Rule for Division

There is an extra rule for division:

As well as restricting the domain as above, when we divide:

(f/g)(x) = f(x) / g(x)

we must also make sure that g(x) is not equal to zero (so we don't divide by zero).

Here is an example:

Example: f(x)=√x and g(x)=√(3-x)

(f/g)(x) = √x / √(3-x)

The domain for f(x)=√x is from 0 onwards:

zero onwards

The domain for g(x)=√(3-x) is up to and including 3:

zero onwards

But we also have the restriction that √(3-x) cannot be zero, so x cannot be 3:

zero onwards
(Notice the open circle at 3, which means not including 3)

So all together we end up with:

zero onwards

 

Summary

  • To add, subtract, multiply or divide functions just do as the operation says.
  • The domain of the new function will have the restrictions of both functions that made it.
  • Divide has the extra rule that the function you are dividing by cannot be zero.

 

 
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