Just as we can perform operations on polynomials, so too can we perform operations on different functions - adding, subtracting, multiplying or dividing them, provided we follow specific rules. The table below defines each of the four operations for functions.
We can perform operations with functions using the following rules:
Operation | Definition | Example using $$f(x)=3x and $$g(x)=2x−1 |
---|---|---|
Addition | $$(f+g)(x)=f(x)+g(x) | $$3x+2x−1=5x−1 |
Subtraction | $$(f−g)(x)=f(x)−g(x) | $$3x−(2x−1)=x+1 |
Multiplication | $$(f·g)(x)=f(x)·g(x) | $$3x(2x−1)=6x2−3x |
Division | $$(fg)(x)=f(x)g(x),g(x)≠0 | $$3x2x−1,x≠12 |
Two functions $$f and $$g may be combined as a sum $$f+g, meaning that for each $$x in the common domain we add the function values $$f(x) and $$g(x) to get $$(f+g)(x). The same can be done with the difference of two functions.
Note that this operation makes no sense unless $$x belongs to the domains of both $$f and $$g. It may be necessary to restrict the domain of one or both functions to meet this requirement.
Let $$f(x)=x2 and $$g(x)=2x+1.
Find $$(f+g)(x) and its domain.
Think: The domains of both functions are the real numbers. So, the sum function $$(f+g)(x) will also have the real numbers for its domain.
Do: We have, $$(f+g)(x)=f(x)+g(x) for each $$x in the domain. Therefore, $$(f+g)(x)=x2+2x+1, and Its domain is the set of real numbers.
We combine functions $$f and $$g as a product $$fg by defining $$(fg)(x)=f(x)·g(x) for each $$x in the common domain of $$f and $$g.
Let $$f(x)=3x and $$g(x)=2x−13.
Find $$(fg)(x) and its domain.
Think: The product function $$(fg)(x) is given by $$f(x)·g(x) over the domain $${R|x≠0}. The domain has to be restricted to the real numbers without zero because this is the domain of $$f.
Do: Therefore, $$(fg)(x)=3x·(2x−13) and so,
$$(fg)(x)=6−1x
We can define a quotient function $$h(x)=f(x)g(x) in a similar way to the way we defined the other operations, provided the domains of $$f and $$g are the same and we do not include in the domain values of $$x that make $$g(x)=0.
Such functions are called rational functions when $$f and $$g are both polynomials.
Given $$f(x)=x2+5x+6 and $$g(x)=2x+1,
Find $$(fg)(x) and its domain.
Think: The quotient function is given by $$(fg)(x)=f(x)g(x), provided that $$g(x)≠0. First, we must find the quotient. Then determine restrictions on its domain by finding the values that make the denominator zero.
Do: Therefore, $$(fg)(x)=x2+5x+62x+1, provided that $$x≠12.
If $$f(x)=3x−5 and $$g(x)=5x+7, find each of the following:
$$(f+g)(x)
$$(f+g)$$(4)
$$(f−g)(x)
$$(f−g)$$(10)
Let $$f(x)=8x3+27 and $$g(x)=2x+3.
What is the domain of $$(f/g)(x)? Give your answer in interval notation.
Simplify the function $$(f/g)(x):
The financial team at The Gamgee Cooperative wants to calculate the profit, $$P(x), generated by producing $$x units of wetsuits.
The revenue produced by the product is given by the equation is $$R(x)=−x24+40x. The cost of production is given by the equation $$C(x)=5x+410.
The profit is calculated as $$P(x)=R(x)−C(x).
Find an equation for $$P(x).
Find the values of the following:
$$R(70)$$=$$
$$C(70)$$=$$
$$P(70)$$=$$
Which of the following correctly displays the graphs of $$y=R(x), $$y=C(x) and $$y=P(x)?
The graph of $$P(x) is represented by the black line in each option.