Chapter 1.4 Combinations of Functions

This chapter covers arithmetic combinations of functions.  This includes the sum, difference, product, and quotient of functions. 


*Note: in order for two functions to be combined they must have an overlapping domain

Sum of Two Functions 

The sum of two functions, f(x) and g(x), can be expressed as follows:

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

Ex. Find the sum of the two functions, f(x)= 2x+1 and g(x)= 9x+11

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

(f+g)(x)= 2x+1+9x+11

(f+g)(x)=11x+12

When graphing the sum of two functions the domain remains the same as the original functions but the y values are that of the two functions added together for each value for x.

Example: the combination of the y values on the blue graph and the y values of the gray graph, at any value of x, yield the y values for red graph. This concept can be applied to all graphs for arithmetic combinations of functions.  

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Difference of Two Functions

The difference of two functions, f(x) and g(x), can be expressed as follows: 

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

Ex. Find the difference of two functions f(x)=13x-5 and g(x)=7x+6

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

(f-g)(x)= 13x-5-(7x+6)   

(f-g)(x)= 13x-5-7x-6

(f-g)(x)= 4x-11

*Always remember to include parenthesis and distribute the negative to all terms when subtracting the second function!!!

When graphing the difference of two functions the domain remains the same as the original functions but the y values are that of the second function subtracted from the first function for any value of x.

Product of Two Functions

The product of two functions, f(x) and g(x), can be expressed as follows: 

(fg)(x)= f(x)g(x)

Ex. Find the product of the two functions f(x)=27x + 1 and g(x)= x+2

(fg)(x)= f(x)g(x)

(fg)(x)= (27x+ 1)(x+2)

When graphing the product of two functions the domain remains the same as the original functions but the y values are that of the y values for each function multiplied together for any value of x.  

Quotient of Two Functions

The quotient of two functions, f(x) and g(x), can be expressed as follows: 

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

Ex. Find the quotient of the two functions and

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

Domain: [0, ∞)

*Note: Often when providing the equation for the quotient of two functions it is often asked that the domain be specified for the function to provide all real solutions 

When graphing the quotient of two functions the domain remains the same as the original functions but the y values are that of the y values for each function divided by one another for any value of x.