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Combinations of Functions

Combinations of Functions

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Author: Eric Bran
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  • Combine functions by adding, subtracting, multiplying, and dividing. 
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Tutorial

The Algebra of Functions

We can combine functions using addition, subtraction, multiplication, and division. Here are examples of what that may look like. 

Let

f left parenthesis x right parenthesis equals 2 x space space space a n d space space space g left parenthesis x right parenthesis equals left parenthesis x minus 1 right parenthesis

The sum f + g looks like :

 left parenthesis f plus g right parenthesis left parenthesis x right parenthesis space equals space f left parenthesis x right parenthesis space plus thin space f left parenthesis g right parenthesis
space space space space space space space space space space space space space space space space equals space 2 x space plus space left parenthesis x minus 1 right parenthesis space space equals space 3 x minus 1

The difference f - g looks like:

left parenthesis f minus g right parenthesis left parenthesis x right parenthesis equals f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis
space space space space space space space space space space space space space space space equals space 2 x minus left parenthesis x minus 1 right parenthesis equals 2 x minus x plus 1 equals x plus 1

The product fg looks like:

left parenthesis f g right parenthesis left parenthesis x right parenthesis equals f left parenthesis x right parenthesis times g left parenthesis x right parenthesis
space space space space space space space space space space space equals 2 x left parenthesis x minus 1 right parenthesis equals 2 x squared minus 2 x

The quotient f/g looks like:

left parenthesis f over g right parenthesis left parenthesis x right parenthesis equals fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction
space space space space space space space space space space space space space equals fraction numerator 2 x over denominator x minus 1 end fraction comma space x not equal to 1.

 

The domain for each of these functions consists of all real numbers that are the common domains of f and g.  The only exception is the quotient of two functions, you must remember that the denominator cannot be zero!

Source: Blitzer - PreCalculus 4e

Finding the sum of two functions.

An example on how to find the difference of two functions.

Finding the difference of two functions.

An example on how to find the difference of two functions.

EDIT: I made a mistake when writing the answer, can you spot it?

Finding the product of two functions.

An example on how to find the product of two functions.

Finding the quotient of two functions

An example on how to find the quotient of two functions.

Individual Practice

F i n d space f plus g comma space f minus g comma space f g space space comma a n d space space f over g. space

1. space f left parenthesis x right parenthesis equals 2 x comma space g left parenthesis x right parenthesis equals x plus 7
2. space f left parenthesis x right parenthesis equals x plus 4 comma space g left parenthesis x right parenthesis equals 2 x plus 1
3. space f left parenthesis x right parenthesis equals 4 x minus 3 comma space g left parenthesis x right parenthesis equals 5 x squared minus 2
4. space f left parenthesis x right parenthesis equals x squared plus 2 comma space g left parenthesis x right parenthesis space equals x squared minus 2
5. space f left parenthesis x right parenthesis equals 4 minus x comma space g left parenthesis x right parenthesis equals 2 x squared plus x plus 5