Operations with Functions (2024)

Operations with Functions (1) Operations with Functions (2)
Operations with Functions (3) Operations with Functions (4)

We can add, subtract, multiply and divide functions!

The result is a new function.

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

Operations with Functions (5)

Addition

We can add two functions:

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

Note: we put the f+g inside () to show they both work on x.

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

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

Sometimes we 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 we will talk about that later!

Operations with Functions (6)

Subtraction

We 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)

Operations with Functions (7)

Multiplication

We 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

Operations with Functions (8)

Division

And we 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)

Domains

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

Operations with Functions (9)

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

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

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

We can't have the square root of a negative number (unless we 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:

Operations with Functions (10)

Using set-builder notation it is written:

{ xOperations with Functions (11)Operations with Functions (12) | x ≥ 0}

"the set of all x's that are a member of the Real Numbers,
such that x is greater than or equal to zero
"

Or using interval notation it is:

[0,+∞)

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

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

How to Work Out the New Domain

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

Operations with Functions (13)

It is like cooking for friends:

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

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

Example: f(x)=√x and g(x)=√(3−x)

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

Operations with Functions (14)

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

Operations with Functions (15)

So the new domain (after adding or whatever) is from 0 to 3:

Operations with Functions (16)

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

In other words we 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) = { xOperations with Functions (17)Operations with Functions (18) | xOperations with Functions (19)Dom(f) and xOperations with Functions (20)Dom(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" (Operations with Functions (21)means "member of")

The same rule applies when we 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)

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

Operations with Functions (22)

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

Operations with Functions (23)

3. AND √(3−x) cannot be zero, so x cannot be 3:

Operations with Functions (24)
(Notice the open circle at 3, which means not including 3)

So all together we end up with:

Operations with Functions (25)

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 we are dividing by cannot be zero.

1193, 1194, 2459, 2460, 8461, 559, 560, 2461, 2462, 8462

Function Composition Algebra Index

Operations with Functions (2024)

References

Top Articles
Latest Posts
Article information

Author: Maia Crooks Jr

Last Updated:

Views: 6278

Rating: 4.2 / 5 (43 voted)

Reviews: 82% of readers found this page helpful

Author information

Name: Maia Crooks Jr

Birthday: 1997-09-21

Address: 93119 Joseph Street, Peggyfurt, NC 11582

Phone: +2983088926881

Job: Principal Design Liaison

Hobby: Web surfing, Skiing, role-playing games, Sketching, Polo, Sewing, Genealogy

Introduction: My name is Maia Crooks Jr, I am a homely, joyous, shiny, successful, hilarious, thoughtful, joyous person who loves writing and wants to share my knowledge and understanding with you.