Algebra (all content)
- Intro to composing functions
- Intro to composing functions
- Composing functions
- Evaluating composite functions
- Evaluate composite functions
- Evaluating composite functions: using tables
- Evaluating composite functions: using graphs
- Evaluate composite functions: graphs & tables
- Finding composite functions
- Find composite functions
- Evaluating composite functions (advanced)
Intro to composing functions
Learn why we'd want to compose two functions together by looking at a farming example.
Cam is a farmer. Each year he plants seeds that turn into corn. The function below gives the amount of corn, , in kilograms (kg), that he expects to produce if he plants corn on acres of land.
For example, if Cam plants two, he expects to produce of corn.
What Cam really wants to know is how much money he will make from selling this corn. So he uses the following function to predict the amount of money, , in dollars, that he will earn from selling kilograms of corn.
So if Cam produces of corn, he can expect to make .
Notice that Cam has to use two separate functions to get from acres planted to expected earnings. The first function, , takes acres to corn, while the second function, , takes corn to money.
Wouldn't it be great if Cam could write a function that turned planted acres directly into expected earnings?
Creating a new function
We can indeed find the function that takes acres planted directly to expected earnings! To find this new function, let's think about the most general question: how much money does Cam expect to make if he plants corn seed on acres of land?
Well, if Cam plants corn on acres, he expects to produce kilograms of corn. And if he produces kilograms of corn, he expects to make dollars.
So, to find a general rule that converts acres directly into expected earnings, we can find the expression .
But just how do we do this? Well, notice that in the expression , the input of function is . So, to find this expression, we can substitute in for in function .
So the function converts acres planted directly into expected earnings. Let's use this new function to predict the amount of money that Cam would make from planting corn on two acres.
Cam can expect to make from planting corn on two acres of land, which is consistent with our previous work!
Defining composite functions
We just found what is called a composite function. Instead of substituting acres planted into the corn function, and then substituting the amount of corn produced into the money function, we found a function that takes the acres planted directly to the expected earnings.
We did this by substituting into function , or by finding . Let's call this new function , which is read as " composed with ".
We now know that . This, in fact, is the formal definition of function composition!
Visualizing the two methods
Here's a visual to help interpret the above definition.
Using both functions and , function —the corn function—takes two to 13,500. Then, function —the money function—takes 13,500 to 12,100.
Using the composite function, we see that function takes two directly to 12,100.
The two are equivalent!
Now let's practice some problems.
Using the functions presented in the example, how much can Cam expect to earn if he sells all the corn produced on 1.5 acres?
For reference: , and
Ben is a potato farmer. The function gives the amount of potatoes, , in kilograms, that he expects to produce from planting potatoes on acres of land. The function gives the amount of money, , in dollars, that Ben expects to make if he produces kilograms of potatoes.
How much money can Ben expect to make if he sells all of the potatoes produced on the 3 acres?
Which of the following expressions gives the amount of money that Ben expects to make if he plants potatoes on acres of land?
Want to join the conversation?
- where did the 1500 come from?(25 votes)
- It is most likely the average expected loss of crops in kg when harvesting.(33 votes)
- Could someone please explain where 6750a came from in Problem One? How was that number found?(16 votes)
- The problem gave you: M(C(a))=6750a−1400
This was created by combining the 2 functions C(a) and M(c) by making C(a) as the input to M(c). Here's how that was done...
We were also given:
M(c) = 0.9c - 50
Insert C(a) as the input into M(c) and here's what M(C(a)) looks like before simplifying:
After you simplify, you get M(C(a)) = 6750a−1400
Hope this helps.(32 votes)
- How would you find the value of the function if like you had f(g(-1)) how would you put that into in equation to solve?(4 votes)
- Since this is currently real world problems, having a negative amount of land is impossible. You would solve it the same way though such as the potato farmer problem by solving P of -1, or substituting it at the end.(1 vote)
- can i get some help with this its kinda getting confusing?(5 votes)
- How to find if a composite function is valid or not? How to find its range and domain?(2 votes)
- How do you find the domain of a composite function?(3 votes)
- The domain of a composite function f(g(x)) is all x in the domain of g such that g(x) is in the domain of f.
Let's break this down. First off, the x has to be in the domain of g; if g(x) were say 1/x, then x = 0 could not be in the composite domain. Second of all, even if g(x) is defined, it has to be in the domain of f. Say f(x) equals 1 / (x - 1). Then if you choose an x such that g(x) = 1, making f(g(x)) = 1 / 0, that x cannot be in the domain of the composite function. Hope that I helped.(3 votes)
- In defining composite functions paragraph 3 it says (M*C)(a) = M(C(a)). Isn't that just multiplying functions? If it says (M*C)(a) why can't I just multiply the two functions?(2 votes)
- Same answer as your other question. Composite function uses an open circle/dot, not a solid dot like multiplication.
Composite: (M o C)(a)
Multiplication: (M * C)(a) or (M • C)(a)(3 votes)
- Problem 1
Shouldn't you solve for C(1.5), then input that value into M(C(a)) rather than just use M(1.5)??
For example, here's my work:
C(1.5) = 7500(1.5) - 1500
11250-1500 = 9750
M(9750) = 6750(9750) - 1400
65812500-1400 = 65811100
I realize that the solution I came up with is unrealistic, but my method of solving seems to me to follow the method taught. So, my question is: why don't you solve it the way I did?(3 votes)
- m(c(a)) = 0.9c _ 50(1 vote)
- If i want to turn the Q around, how am i suppose to do it. Do i just sub the function around?(2 votes)
- What do I do if I have to find f(x)h(x)?(1 vote)
- Multiply the two functions. so say f(x) = 5x and g(x) = 3x^2 then f(x)g(x) = 5x*3x^2 = 5*5*x*x^2 = 15x^3 Does that make sense?
Changed 15x^2 to 15x^3, thanks to Mr. K for pointing it out.(3 votes)
- I don't understand problem 3 can anyone explain?(1 vote)
- This is using the same logic as Cam and his carrot planting where with Ben for example how you would of solved problem 2 is solving how much carrots he would of planted on 3 acres of land with P(a) = 25,000a - 1000 But here we have P(3) = 25,000(3) - 1000 and we have 75,000 - 1000 which gives us 74,000 potatoes. Now we want to see how much money he can make with all those potatoes so we put that into the second equation/function M(p) -notice that the p is the amount of potatoes from P(a)
M(p) = 0.2p - 200 But here we have M(74,000) = 0.2(74,000) - 200 which gives us
14,800 - 200 and we finally have 14,600 or in this case $14,600
But instead if switching the answer of one function into another we can compose or combine the two like since we want to find the amount of money from 3 acres of land we can start with M(p) but we can use the potatoes function as the input like M( p(a) ) but note that this is still basically going back and forth with the both of them but now we can input it into the function M to get M( p(a) ) = 0.2 ( p(a) - 200 and p(a) is 25,000a - 1000 right? So now we can use that and get M ( p(a) ) = 0.2 ( 25,000a - 1000 ) - 200
Now we can expand the parenthesis and finally get M ( p(a) ) = 5000a - 200 - 200 which we can combine the like terms to get M ( p(a) ) = 5000a - 400
When you put 3 into that you have 15,000 - 400 which is 14,600 or also $14,600
-Note you could put the function P equation in the M domain slot but it easier to write it as P(a) and easier to read and understand what it is.
--Hope this helps :)(2 votes)