Recall that (f± g)(x)=f(x)± g(x), (f* g)(x)=f(x)* g(x), and ( fg)(x)= f(x)g(x). Then perform the function operations.
Function Operation
Result
(f+g)(x)
4x+1
(f-g)(x)
- 2x+3
(f* g)(x)
3x^2+5x-2
(f/g)(x)
x+2/3x-1, x ≠1/3
Practice makes perfect
We want to perform arithmetic operations with the functions f(x) and g(x). To do so, we will calculate (f+g)(x), (f-g)(x), (f* g)(x), and ( fg)(x) separately.
Calculating (f+g)(x)
When we are given the addition of two functions written as (f+g)(x), we can think of this as f(x)+g(x).
(f+g)(x) ⇔ f(x)+g(x)
Let's substitute the given function rules into the expressions f(x) and g(x). Then we will perform the addition.
After we have performed the addition, the result takes the form of 4x+1.
(f+g)(x) = 4x+1
Calculating (f-g)(x)
When we are given the subtraction of two functions written as (f-g)(x), we can think of this as f(x)-g(x).
(f-g)(x) ⇔ f(x)-g(x)
Let's substitute the given function rules into the expressions f(x) and g(x). Then we will perform the subtraction.
After we have performed the subtraction, the result takes the form of - 2x+3.
(f-g)(x) = - 2x+3
Calculating (f* g)(x)
When we are given the multiplication of two functions written as (f* g)(x), we can think of this as f(x)* g(x).
(f* g)(x) ⇔ f(x)* g(x)
Let's substitute the given function rules into the expressions f(x) and g(x). Then we will perform the multiplication.
After we have performed the multiplication, the result takes the form of 3x^2+5x-2.
(f* g)(x) = 3x^2+5x-2
Calculating ( fg)(x)
When we are given the division of two functions written as ( fg)(x). We can think of this as f(x)g(x).
(f/g)(x) ⇔ f(x)/g(x)
To perform the division, let's substitute the given function rules into the expressions f(x) and g(x).
This expression cannot be simplified any further. The domain of both f(x) and g(x) is all real numbers, R. The domain of f(x)g(x) is the set of numbers common to the domains of both f(x) and g(x). However, the denominator cannot be 0. Let's find the values that make g(x)=0.
As we can see, x= 13 makes g(x)=0. Therefore, this value must be excluded from the domain of the quotient. We can say that the domain is the set of all real numbers except 13.
(f/g)(x) = x+2/3x-1, x ≠1/3
