In practice Q 4, where is 4t created? I see where t^2 and 4 come from, but am not sure what puts 4t in

I was stuck on this too, but I think the reason is that (t-2)^2 = (t-2)(t-2) . Using the distributive property, you get t^2-4t+4.

How do you know when to use the "inside out property" or the composing function?

It doesn't really matter --- they will both give the same answer, so it's up to you to choose what works best/easiest for you with the problem you're given at the time! (But, of course, you need to be familiar with both techniques.)

If f(x)=(1/x) and (f/g)(x)=((x+4)/(x^2+2x)), what is the function g?

Based upon the rules for dividing with fractions: f/g = (1/x) / g = (1/x) * the reciprocal of g We need to work in reverse 1) Factor denominator to undo the multiplication: `(x+4)/(x^2+2x)` = `(x+4)/[x(x+2)]` We can see there is a factor of X in the denominator. This would have been from multiplying 1/x * the reciprocal of g. 2) Separate the factor 1/x: `(1/x) * (x+4)/(x+2)` This tells us the reciprocal of g = `(x+4)/(x+2)` 3) Flip it to find g: `g(x) = (x+2)/(x+4)` Hope this helps.

In problem 4, i got t^2 +9, so where did the -4t come from? I tried to find out back in the examples but it did the same. Its like taking the number inside the parenthesis and multiplying it by 2t or 2 times the variable. =(x+4)^2−2(x+4) =x^2+8x+16−2x−8...this was the example, so where did 8x come from?

Your mistake was simple. You didn't go about the exponent property the correct way. This is how I solved it: (gof)(t)--->g(f(t)) g(ft)=(ft)^2+5...remember ft=(t-2) g(t-2)=(t-2)^2+5 at this point (t-2)^2, is where you made the error. I'm sure you probably distributed the 2 instead of multiplying (t-2) twice. But instead, FOIL it, and you get the quadratic function: t^2-4t+4. Hence, where "-4t" came from. (:

(f ∘ g)(x) here, what does the sign ∘ mean?

(f ∘ g)(x) is read "f of g of x", so the ∘ translates to "of". In this case, if you had functions defined, f(x) and g(x), then to get (f ∘ g)(x) you would substitute g(x) for x inside of f(x). Another way to write it is f(g(x)).

May someone please explain the challenge problem to me?

The challenge problem says, "The graphs of the equations y=f(x) and y=g(x) are shown in the grid below." So basically the two graphs is a visual representation of what the two different functions would look like if graphed and they're asking us to find (f∘g)(8), which is combining the two functions and inputting 8. From the definition, we know (f∘g)(8)=f(g(8)). So let's work "inside out". If we look at the graph of "g", we see that g(8) is 2 (look at the 8 at the x-axis and if you go up to where it meets the line, the y value would be 2). Because g(8)=2, then when you substitute it back in the equation, f(g(8)) would equal f(2). Then if we look at the graph of "f", we can see that f(2) is -3. (when you look at the 2 in the x-axis, it will correspond to -3 on the y-axis). So by looking at the graph, you can figure out that (f∘g)(8) is approximately -3. ~Dylan

Can someone please simplify all of this for me cause i am so confused!

Sometimes it's useful to look at a different point of view. Try this site. Then come back and try this video again. http://www.mathsisfun.com/sets/functions-composition.html

In the last question, is there a problem with the graph because, I'm finding it difficult to understand it?

The graph is fine, but this question pools together some concepts from other parts of math the we may have forgotten about. `f(x)` is your standard linear equation of the form ```f(x) = mx + b``` Specifically it's ```f(x) = -0.5x - 2``` g(x) is a sinusoidal equation of the form ```g(x) = a cos(b (x - c)) + d``` Where `a` is the amplitude, `2π/|b|` is the period, `c` is the phase shift, and `d` is the midline. Finding `a` is simple enough; it's `2`. The phase shift and midline both look like `0`, so that's nice. The tricky part is finding `b`. The period of this particular sinusoidal function appears to be `8`, which means we need to solve for `b` where ``` 2π/|b| = 8 # Do some algebra... b = π/4 ``` Once we have done this we can compose the functions like so ``` f(x) = -0.5 * x - 2 g(x) = 2 * cos(π / 4 * x) f(g(x)) = -0.5 * (2 * cos(π / 4 * x)) - 2 ``` Which we can further simplify as ``` f(g(x)) = -cos(π / 4 * x) - 2 ``` Of course you could forgo composing the functions and evaluate each function separately. Either way, you'll find that `f(g(x)) ~= -3`. Here's a graph of everything: https://www.desmos.com/calculator/b0r64jqiqv

what if its f o f (x)? will it be f(x)^2? i heard that there's a _definition_ that states that *f o f o f ....(n times)= f(x)^n* is it true?

In some contexts, composing f n times is written as fⁿ(x), but we don't usually use this notation in K-12 or on Khan Academy. f²(x) usually means the product f(x)·f(x). sin²(x) is used this way.

For the challenge problem, since g(8)=2 why does f(g(8)) also equal f(2)? That part just really confused me

Because g(8) = 2, we can substitute 2 for g(8) anywhere it appears. That's what has happened in f(g(8)) to get f(2).

Main content

Composing functions

CCSS.Math:

HSF.BF.A.1c

Google Classroom

Walk through examples, explanations, and practice problems to learn how to find and evaluate composite functions.

Given two functions, we can combine them in such a way so that the outputs of one function become the inputs of the other. This action defines a composite function. Let's take a look at what this means!

Evaluating composite functions

Example

If f, left parenthesis, x, right parenthesis, equals, 3, x, minus, 1 and g, left parenthesis, x, right parenthesis, equals, x, cubed, plus, 2, then what is f, left parenthesis, g, left parenthesis, 3, right parenthesis, right parenthesis?

Solution

One way to evaluate f, left parenthesis, g, left parenthesis, 3, right parenthesis, right parenthesis is to work from the "inside out". In other words, let's evaluate g, left parenthesis, 3, right parenthesis first and then substitute that result into f to find our answer.

Let's evaluate g, left parenthesis, 3, right parenthesis.

$\begin{aligned}g(x)&=x^3+2\\\\ g(3)&=({3})^3 +2~~~~~~~~~~~~~~~~~~~\small{\gray{\text{Plug in }x={3.}}}\\\\ &={29}\end{aligned}$

Since g, left parenthesis, 3, right parenthesis, equals, 29, then f, left parenthesis, g, left parenthesis, 3, right parenthesis, right parenthesis, equals, f, left parenthesis, 29, right parenthesis.

Now let's evaluate f, left parenthesis, 29, right parenthesis.

$\begin{aligned}f(x)&=3x-1\\\\ f( {{29}})&=3({29}) - 1~~~~~~~~~~~~~~~\small{\gray{\text{Plug in }x= {29.}}}\\\\ &={86}\\\\ \end{aligned}$

It follows that f, left parenthesis, g, left parenthesis, 3, right parenthesis, right parenthesis, equals, f, left parenthesis, 29, right parenthesis, equals, 86.

Finding the composite function

In the above example, function g took 3 to 29, and then function f took 29 to 86. Let's find the function that takes 3 directly to 86.

To do this, we must compose the two functions and find f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis.

Example

What is f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis?
For reference, remember that f, left parenthesis, x, right parenthesis, equals, 3, x, minus, 1 and g, left parenthesis, x, right parenthesis, equals, x, cubed, plus, 2.

Solution

If we look at the expressionf, left parenthesis, start color #ca337c, g, left parenthesis, x, right parenthesis, end color #ca337c, right parenthesis, we can see that start color #ca337c, g, left parenthesis, x, right parenthesis, end color #ca337c is the input of function f. So, let's substitute start color #ca337c, g, left parenthesis, x, right parenthesis, end color #ca337c everywhere we see start color #0c7f99, x, end color #0c7f99 in function f.

$\begin{aligned}f(\blueE x)&=3\blueE x-1\\\\ f(\maroonD{g(x)}) &= 3(\maroonD{g(x)})-1 \\ \end{aligned}$

Since g, left parenthesis, x, right parenthesis, equals, x, cubed, plus, 2, we can substitute x, cubed, plus, 2 in for g, left parenthesis, x, right parenthesis.

$\begin{aligned}{f(g(x))}&=3(g(x))-1 \\\\ &=3({x^3+2})-1 \\\\ &=3x^3+6-1\\\\ &=3x^3+5 \end{aligned}$

This new function should take 3 directly to 86. Let's verify this.

$\begin{aligned} f( g(x))&= 3x^3+5\\ \\ f( g( 3))&= 3( 3)^3+5 \\\\ &= {86} \end{aligned}$

Excellent!

Let's practice

Problem 1

f, left parenthesis, x, right parenthesis, equals, 3, x, minus, 1

g, left parenthesis, x, right parenthesis, equals, x, cubed, plus, 2

Evaluate g, left parenthesis, f, left parenthesis, 1, right parenthesis, right parenthesis.

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

Problem 2

m, left parenthesis, x, right parenthesis, equals, 3, x, minus, 2

n, left parenthesis, x, right parenthesis, equals, x, plus, 4

Find m, left parenthesis, n, left parenthesis, x, right parenthesis, right parenthesis.

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

Composite functions: a formal definition

In the above example, we found and evaluated a composite function.

In general, to indicate function f composed with function g, we can write f, circle, g, read as "f composed with g". This composition is defined by the following rule:

left parenthesis, f, circle, g, right parenthesis, left parenthesis, x, right parenthesis, equals, f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis

The diagram below shows the relationship between left parenthesis, f, circle, g, right parenthesis, left parenthesis, x, right parenthesis and f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis.

Now let's look at another example with this new definition in mind.

Example

g, left parenthesis, x, right parenthesis, equals, x, plus, 4

h, left parenthesis, x, right parenthesis, equals, x, squared, minus, 2, x

Find left parenthesis, h, circle, g, right parenthesis, left parenthesis, x, right parenthesis and left parenthesis, h, circle, g, right parenthesis, left parenthesis, minus, 2, right parenthesis.

Solution

We can find left parenthesis, h, circle, g, right parenthesis, left parenthesis, x, right parenthesis as follows:

$\begin{aligned}(h\circ g)(x)&=h(g(x))&\small{\gray{\text{Define.}}}\\\\ &=(g(x))^2-2(g(x))&\small{\gray{\text{Plug } g(x) \text{ in for } x\text{ in function }h.}}\\\\ &=({x+4})^2 -2({x+4})&\small{\gray{\text{Substitute } x+4 \text{ for } g(x).}}\\\\ &=x^2+8x+16-2x-8&\small{\gray{\text{Distribute.}}}\\\\ &=x^2+6x+8&\small{\gray{\text{Combine like terms.}}}\end{aligned}$

Since we now have function h, circle, g, we can simply substitute minus, 2 in for x to find left parenthesis, h, circle, g, right parenthesis, left parenthesis, minus, 2, right parenthesis.

$\begin{aligned}(h\circ g)(x)&=x^2+6x+8\\\\ (h\circ g)(-2)&=(-2)^2+6(-2)+8\\\\ &=4-12+8\\\\ &=0\\\\ \end{aligned}$

Of course, we could have also found left parenthesis, h, circle, g, right parenthesis, left parenthesis, minus, 2, right parenthesis by evaluating h, left parenthesis, g, left parenthesis, minus, 2, right parenthesis, right parenthesis. This is shown below:

$\begin{aligned}(h\circ g)(-2)&=h(g(-2))\\\\ &=h(2)~~~~~~~~\small{\gray{\text{Since }g(-2)=-2+4=2}}\\\\ &=0~~~~~~~~~~~~~\small{\gray{\text{Since }h(2)=2^2-2(2)=0}}\\\\ \end{aligned}$

The diagram below shows how left parenthesis, h, circle, g, right parenthesis, left parenthesis, minus, 2, right parenthesis is related to h, left parenthesis, g, left parenthesis, minus, 2, right parenthesis, right parenthesis.

Here we can see that function g takes minus, 2 to 2 and then function h takes 2 to 0, while function h, circle, g takes minus, 2 directly to 0.

Now let's practice some problems

Problem 3

f, left parenthesis, x, right parenthesis, equals, 3, x, minus, 5

g, left parenthesis, x, right parenthesis, equals, 3, minus, 2, x

Evaluate left parenthesis, g, circle, f, right parenthesis, left parenthesis, 3, right parenthesis.

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

In problems 4 and 5, let f, left parenthesis, t, right parenthesis, equals, t, minus, 2 and g, left parenthesis, t, right parenthesis, equals, t, squared, plus, 5.

Problem 4

Find left parenthesis, g, circle, f, right parenthesis, left parenthesis, t, right parenthesis.

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

Problem 5

Find left parenthesis, f, circle, g, right parenthesis, left parenthesis, t, right parenthesis.

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

Challenge Problem

The graphs of the equations y, equals, f, left parenthesis, x, right parenthesis and y, equals, g, left parenthesis, x, right parenthesis are shown in the grid below.

Which of the following best approximates the value of left parenthesis, f, circle, g, right parenthesis, left parenthesis, 8, right parenthesis?

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

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Sort by:

Tess Van Horn
Posted 7 years ago. Direct link to Tess Van Horn's post “In practice Q 4, where is...”
In practice Q 4, where is 4t created? I see where t^2 and 4 come from, but am not sure what puts 4t in
Button navigates to signup pageComment on Tess Van Horn's post “In practice Q 4, where is...”
(55 votes)
Answer
- Mr.Magroo
  Posted 7 years ago. Direct link to Mr.Magroo's post “I was stuck on this too, ...”
  I was stuck on this too, but I think the reason is that (t-2)^2 = (t-2)(t-2) . Using the distributive property, you get t^2-4t+4.
  Comment on Mr.Magroo's post “I was stuck on this too, ...”
  (95 votes)
Nigar Kainath
Posted 7 years ago. Direct link to Nigar Kainath's post “(f ∘ g)(x) here, what doe...”
(f ∘ g)(x)
here, what does the sign ∘ mean?
Button navigates to signup pageComment on Nigar Kainath's post “(f ∘ g)(x) here, what doe...”
(2 votes)
Answer
- Levi Geadelmann
  Posted 7 years ago. Direct link to Levi Geadelmann's post “(f ∘ g)(x) is read "f of ...”
  (f ∘ g)(x) is read "f of g of x", so the ∘ translates to "of".
  In this case, if you had functions defined, f(x) and g(x), then to get (f ∘ g)(x) you would substitute g(x) for x inside of f(x). Another way to write it is f(g(x)).
  Button navigates to signup page
  (10 votes)
Rory Avera
Posted 6 years ago. Direct link to Rory Avera's post “How do you know when to u...”
How do you know when to use the "inside out property" or the composing function?
Button navigates to signup pageComment on Rory Avera's post “How do you know when to u...”
(8 votes)
Answer
- Judith Gibson
  Posted 6 years ago. Direct link to Judith Gibson's post “It doesn't really matter ...”
  It doesn't really matter --- they will both give the same answer, so it's up to you to choose what works best/easiest for you with the problem you're given at the time!
  (But, of course, you need to be familiar with both techniques.)
  Button navigates to signup page
  (6 votes)
Aditya Mahajan
Posted 4 years ago. Direct link to Aditya Mahajan's post “May someone please explai...”
May someone please explain the challenge problem to me?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Dylan Chan
  Posted 4 years ago. Direct link to Dylan Chan's post “The challenge problem say...”
  The challenge problem says, "The graphs of the equations y=f(x) and y=g(x) are shown in the grid below." So basically the two graphs is a visual representation of what the two different functions would look like if graphed and they're asking us to find (f∘g)(8), which is combining the two functions and inputting 8. From the definition, we know (f∘g)(8)=f(g(8)). So let's work "inside out". If we look at the graph of "g", we see that g(8) is 2 (look at the 8 at the x-axis and if you go up to where it meets the line, the y value would be 2). Because g(8)=2, then when you substitute it back in the equation, f(g(8)) would equal f(2). Then if we look at the graph of "f", we can see that f(2) is -3. (when you look at the 2 in the x-axis, it will correspond to -3 on the y-axis). So by looking at the graph, you can figure out that (f∘g)(8) is approximately -3.
  ~Dylan
  Button navigates to signup page
  (12 votes)
awesomeness.RM
Posted 7 years ago. Direct link to awesomeness.RM's post “Can someone please simpli...”
Can someone please simplify all of this for me cause i am so confused!
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Kim Seidel
  Posted 7 years ago. Direct link to Kim Seidel's post “Sometimes it's useful to ...”
  Sometimes it's useful to look at a different point of view. Try this site. Then come back and try this video again. http://www.mathsisfun.com/sets/functions-composition.html
  Comment on Kim Seidel's post “Sometimes it's useful to ...”
  (6 votes)
Jennifer Laessig
Posted 6 years ago. Direct link to Jennifer Laessig's post “If f(x)=(1/x) and (f/g)(x...”
If f(x)=(1/x) and (f/g)(x)=((x+4)/(x^2+2x)), what is the function g?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Kim Seidel
  Posted 6 years ago. Direct link to Kim Seidel's post “Based upon the rules for ...”
  Based upon the rules for dividing with fractions: f/g = (1/x) / g = (1/x) * the reciprocal of g
  
  We need to work in reverse
  1) Factor denominator to undo the multiplication: (x+4)/(x^2+2x) = (x+4)/[x(x+2)]
  We can see there is a factor of X in the denominator. This would have been from multiplying 1/x * the reciprocal of g.
  2) Separate the factor 1/x: (1/x) * (x+4)/(x+2)
  This tells us the reciprocal of g = (x+4)/(x+2)
  
  3) Flip it to find g: g(x) = (x+2)/(x+4)
  
  Hope this helps.
  Button navigates to signup page
  (2 votes)
Jasmine Flores
Posted 7 years ago. Direct link to Jasmine Flores's post “In problem 4, i got t^2 +...”
In problem 4, i got t^2 +9, so where did the -4t come from? I tried to find out back in the examples but it did the same. Its like taking the number inside the parenthesis and multiplying it by 2t or 2 times the variable.
=(x+4)^2−2(x+4) =x^2+8x+16−2x−8...this was the example, so where did 8x come from?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Yvette Sanders
  Posted 7 years ago. Direct link to Yvette Sanders's post “Your mistake was simple. ...”
  Your mistake was simple. You didn't go about the exponent property the correct way. This is how I solved it:
  (gof)(t)--->g(f(t))
  g(ft)=(ft)^2+5...remember ft=(t-2)
  g(t-2)=(t-2)^2+5
  at this point (t-2)^2, is where you made the error. I'm sure you probably distributed the 2 instead of multiplying (t-2) twice. But instead, FOIL it, and you get the quadratic function: t^2-4t+4.
  Hence, where "-4t" came from. (:
  Comment on Yvette Sanders's post “Your mistake was simple. ...”
  (2 votes)
Arbaaz Ibrahim
Posted 5 years ago. Direct link to Arbaaz Ibrahim's post “In the last question, is ...”
In the last question, is there a problem with the graph because, I'm finding it difficult to understand it?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- ozbonus
  Posted 3 years ago. Direct link to ozbonus's post “The graph is fine, but th...”
  The graph is fine, but this question pools together some concepts from other parts of math the we may have forgotten about.
  
  f(x) is your standard linear equation of the form
  
  f(x) = mx + b
  
  Specifically it's
  
  f(x) = -0.5x - 2
  
  g(x) is a sinusoidal equation of the form
  
  g(x) = a cos(b (x - c)) + d
  
  Where a is the amplitude, 2π/|b| is the period, c is the phase shift, and d is the midline.
  
  Finding a is simple enough; it's 2. The phase shift and midline both look like 0, so that's nice. The tricky part is finding b.
  
  The period of this particular sinusoidal function appears to be 8, which means we need to solve for b where
  
  2π/|b| = 8 # Do some algebra... b = π/4
  
  Once we have done this we can compose the functions like so
  
  f(x) = -0.5 * x - 2 g(x) = 2 * cos(π / 4 * x) f(g(x)) = -0.5 * (2 * cos(π / 4 * x)) - 2
  
  Which we can further simplify as
  
  f(g(x)) = -cos(π / 4 * x) - 2
  
  Of course you could forgo composing the functions and evaluate each function separately. Either way, you'll find that f(g(x)) ~= -3.
  
  Here's a graph of everything: https://www.desmos.com/calculator/b0r64jqiqv
  Button navigates to signup page
  (4 votes)
Khushi
Posted 6 years ago. Direct link to Khushi's post “For the challenge problem...”
For the challenge problem, since g(8)=2 why does f(g(8)) also equal f(2)? That part just really confused me
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Judith Gibson
  Posted 6 years ago. Direct link to Judith Gibson's post “Because g(8) = 2, we can ...”
  Because g(8) = 2, we can substitute 2 for g(8) anywhere it appears.
  That's what has happened in f(g(8)) to get f(2).
  Comment on Judith Gibson's post “Because g(8) = 2, we can ...”
  (4 votes)
bh
Posted 4 years ago. Direct link to bh's post “what if its f o f (x)? wi...”
what if its f o f (x)? will it be f(x)^2?
i heard that there's a definition that states that
f o f o f ....(n times)= f(x)^n
is it true?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- kubleeka
  Posted 4 years ago. Direct link to kubleeka's post “In some contexts, composi...”
  In some contexts, composing f n times is written as fⁿ(x), but we don't usually use this notation in K-12 or on Khan Academy. f²(x) usually means the product f(x)·f(x). sin²(x) is used this way.
  Button navigates to signup page
  (2 votes)

Algebra (all content)

Course: Algebra (all content) > Unit 7

Composing functions

Evaluating composite functions

Example

Solution

Finding the composite function

Example

Solution

Let's practice

Problem 1

Problem 2

Composite functions: a formal definition

Example

Solution

Now let's practice some problems

Problem 3

Problem 4

Problem 5

Challenge Problem

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