where did the 1500 come from?

It is most likely the average expected loss of crops in kg when harvesting.

Could someone please explain where 6750a came from in Problem One? How was that number found?

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: C(a)=7500a−1500 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: M(C(a))=0.9(7500a−1500)−50 After you simplify, you get M(C(a)) = 6750a−1400 Hope this helps.

can i get some help with this its kinda getting confusing?

How to find if a composite function is valid or not? How to find its range and domain?

How do you find the domain of a composite function?

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.

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?

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)

What do I do if I have to find f(x)h(x)?

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? EDIT Changed 15x^2 to 15x^3, thanks to Mr. K for pointing it out.

I don't understand problem 3 can anyone explain?

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 :)

Main content

Intro to composing functions

CCSS.Math:

HSF.BF.A.1c

Google Classroom

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, C, in kilograms (kg), that he expects to produce if he plants corn on a acres of land.

C, left parenthesis, a, right parenthesis, equals, 7500, a, minus, 1500

For example, if Cam plants two, he expects to produce C, left parenthesis, 2, right parenthesis, equals, 7500, left parenthesis, 2, right parenthesis, minus, 1500, equals, 13, comma, 500 start text, k, g, end text 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, M, in dollars, that he will earn from selling c kilograms of corn.

M, left parenthesis, c, right parenthesis, equals, 0, point, 9, c, minus, 50

So if Cam produces 13, comma, 500, start text, space, k, g, end text of corn, he can expect to make M, left parenthesis, 13, comma, 500, right parenthesis, equals, 0, point, 9, left parenthesis, 13, comma, 500, right parenthesis, minus, 50, equals, dollar sign, 12, comma, 100.

Notice that Cam has to use two separate functions to get from acres planted to expected earnings. The first function, C, takes acres to corn, while the second function, M, 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 a acres of land?

Well, if Cam plants corn on a acres, he expects to produce C, left parenthesis, a, right parenthesis kilograms of corn. And if he produces C, left parenthesis, a, right parenthesis kilograms of corn, he expects to make M, left parenthesis, C, left parenthesis, a, right parenthesis, right parenthesis dollars.

So, to find a general rule that converts a acres directly into expected earnings, we can find the expression M, left parenthesis, C, left parenthesis, a, right parenthesis, right parenthesis.

But just how do we do this? Well, notice that in the expression M, left parenthesis, start color #1fab54, C, left parenthesis, a, right parenthesis, end color #1fab54, right parenthesis, the input of function M is start color #1fab54, C, left parenthesis, a, right parenthesis, end color #1fab54. So, to find this expression, we can substitute start color #1fab54, C, left parenthesis, a, right parenthesis, end color #1fab54 in for start color #e07d10, c, end color #e07d10 in function M.

$\begin{aligned} M(\goldD c)&=0.9\goldD c-50\\\\ M({\greenD{C(a)}})&=0.9(\greenD{C(a)})-50\\ \\ &= 0.9(\greenD{7500a-1500})-50~~~~~~~~~~\small{\gray{\text{Since }}}\small{\gray{C(a)=7500a-1500} }\\\\ &= 6750a-1350-50\\\\ &=6750a-1400 \end{aligned}$

So the function M, left parenthesis, C, left parenthesis, a, right parenthesis, right parenthesis, equals, 6750, a, minus, 1400 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.

M, left parenthesis, C, left parenthesis, 2, right parenthesis, right parenthesis, equals, 6750, left parenthesis, 2, right parenthesis, minus, 1400, equals, dollar sign, 12, comma, 100

Cam can expect to make dollar sign, 12, comma, 100 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 C, left parenthesis, a, right parenthesis into function M, or by finding M, left parenthesis, C, left parenthesis, a, right parenthesis, right parenthesis. Let's call this new function M, circle, C, which is read as "M composed with C".

We now know that left parenthesis, M, circle, C, right parenthesis, left parenthesis, a, right parenthesis, equals, M, left parenthesis, C, left parenthesis, a, right parenthesis, right parenthesis. 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 C and M, function C—the corn function—takes two to 13,500. Then, function M—the money function—takes 13,500 to dollar sign12,100.

Using the composite function, we see that function M, circle, C takes two directly to dollar sign12,100.

The two are equivalent!

Now let's practice some problems.

Problem 1

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: C, left parenthesis, a, right parenthesis, equals, 7500, a, minus, 1500, M, left parenthesis, c, right parenthesis, equals, 0, point, 9, c, minus, 50 and M, left parenthesis, C, left parenthesis, a, right parenthesis, right parenthesis, equals, 6750, a, minus, 1400

dollars

[I need help. Please show me the solution.]

Problem 2

Ben is a potato farmer. The function P, left parenthesis, a, right parenthesis, equals, 25, comma, 000, a, minus, 1000 gives the amount of potatoes, P, in kilograms, that he expects to produce from planting potatoes on a acres of land. The function M, left parenthesis, p, right parenthesis, equals, 0, point, 2, p, minus, 200 gives the amount of money, M, in dollars, that Ben expects to make if he produces p kilograms of potatoes.

How much money can Ben expect to make if he sells all of the potatoes produced on the 3 acres?

[I need help. Please show me the solution.]

Problem 3

Which of the following expressions gives the amount of money that Ben expects to make if he plants potatoes on a acres of land?

(Choice A)
0, point, 2, a, minus, 200
(Choice B)
5000, a, minus, 400
(Choice C)
left parenthesis, 25, comma, 000, a, minus, 1000, right parenthesis, left parenthesis, 0, point, 2, p, minus, 200, right parenthesis

[I need help. Please show me the solution.]

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ashley santana
Posted 7 years ago. Direct link to ashley santana's post “where did the 1500 come f...”
where did the 1500 come from?
Button navigates to signup pageComment on ashley santana's post “where did the 1500 come f...”
(25 votes)
Answer
- xilverd020
  Posted 6 years ago. Direct link to xilverd020's post “It is most likely the ave...”
  It is most likely the average expected loss of crops in kg when harvesting.
  Comment on xilverd020's post “It is most likely the ave...”
  (31 votes)
Liah C.
Posted 6 years ago. Direct link to Liah C.'s post “Could someone please expl...”
Could someone please explain where 6750a came from in Problem One? How was that number found?
Button navigates to signup pageComment on Liah C.'s post “Could someone please expl...”
(16 votes)
Answer
- Kim Seidel
  Posted 6 years ago. Direct link to Kim Seidel's post “The problem gave you: M(...”
  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:
  C(a)=7500a−1500
  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:
  M(C(a))=0.9(7500a−1500)−50
  After you simplify, you get M(C(a)) = 6750a−1400
  
  Hope this helps.
  Comment on Kim Seidel's post “The problem gave you: M(...”
  (32 votes)
Shalaya
Posted 7 years ago. Direct link to Shalaya's post “How would you find the va...”
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?
Button navigates to signup pageComment on Shalaya's post “How would you find the va...”
(4 votes)
Answer
- Dawson Stokes
  Posted 7 years ago. Direct link to Dawson Stokes's post “Since this is currently r...”
  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.
  Comment on Dawson Stokes's post “Since this is currently r...”
  (1 vote)
Jason Reed
Posted 7 years ago. Direct link to Jason Reed's post “can i get some help with ...”
can i get some help with this its kinda getting confusing?
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(4 votes)
Answer
- anishbanerjee65
  Posted 6 years ago. Direct link to anishbanerjee65's post “How to find if a composit...”
  How to find if a composite function is valid or not? How to find its range and domain?
  Button navigates to signup page
  (2 votes)
Silverleaf.yen
Posted 4 years ago. Direct link to Silverleaf.yen's post “How do you find the domai...”
How do you find the domain of a composite function?
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(3 votes)
Answer
- Alex
  Posted 4 years ago. Direct link to Alex's post “The domain of a composite...”
  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.
  Button navigates to signup page
  (3 votes)
Liam Heraty
Posted 6 years ago. Direct link to Liam Heraty's post “In defining composite fun...”
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?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Kim Seidel
  Posted 6 years ago. Direct link to Kim Seidel's post “Same answer as your other...”
  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)
  Button navigates to signup page
  (3 votes)
Alex Cail
Posted 6 years ago. Direct link to Alex Cail's post “Problem 1 Shouldn't you s...”
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:
1.5 acres
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?
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(3 votes)
Answer
- jesse suarez
  Posted 5 years ago. Direct link to jesse suarez's post “m(c(a)) = 0.9c _ 50”
  m(c(a)) = 0.9c _ 50
  Button navigates to signup page
  (1 vote)
Ace Matei
Posted 3 years ago. Direct link to Ace Matei's post “What do I do if I have to...”
What do I do if I have to find f(x)h(x)?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- loumast17
  Posted 3 years ago. Direct link to loumast17's post “Multiply the two function...”
  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?
  
  EDIT
  
  Changed 15x^2 to 15x^3, thanks to Mr. K for pointing it out.
  Comment on loumast17's post “Multiply the two function...”
  (3 votes)
Helen
Posted 6 years ago. Direct link to Helen's post “I don't understand proble...”
I don't understand problem 3 can anyone explain?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- programminglegend1029
  Posted 6 years ago. Direct link to programminglegend1029's post “This is using the same lo...”
  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 :)
  Button navigates to signup page
  (2 votes)
Evangeline Liew
Posted 5 years ago. Direct link to Evangeline Liew's post “If i want to turn the Q a...”
If i want to turn the Q around, how am i suppose to do it. Do i just sub the function around?
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(1 vote)
Answer

Precalculus

Course: Precalculus > Unit 1

Intro to composing functions

Creating a new function

Defining composite functions

Visualizing the two methods

Now let's practice some problems.

Problem 1

Problem 2

Problem 3

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