If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Composing functions

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.
g(x)=x3+2g(3)=(3)3+2                   Plug in x=3.=29\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.
f(x)=3x1f(29)=3(29)1               Plug in x=29.=86\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.
f(x)=3x1f(g(x))=3(g(x))1\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.
f(g(x))=3(g(x))1=3(x3+2)1=3x3+61=3x3+5\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.
f(g(x))=3x3+5f(g(3))=3(3)3+5=86\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.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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.

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:
(hg)(x)=h(g(x))Define.=(g(x))22(g(x))Plug g(x) in for x in function h.=(x+4)22(x+4)Substitute x+4 for g(x).=x2+8x+162x8Distribute.=x2+6x+8Combine like terms.\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.
(hg)(x)=x2+6x+8(hg)(2)=(2)2+6(2)+8=412+8=0\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:
(hg)(2)=h(g(2))=h(2)        Since g(2)=2+4=2=0             Since h(2)=222(2)=0\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.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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.

Problem 5

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

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?
Choose 1 answer:
Choose 1 answer:

Want to join the conversation?

  • blobby green style avatar for user Tess Van Horn
    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
    (53 votes)
    Default Khan Academy avatar avatar for user
    • leaf red style avatar for user Paul
      Actually, where 4t was created was based of a theorem. (a-b)^2 = a^2 + b^2 - 2ab.
      let a be t and b be 2 to get: t^2+2-2(-2)t. Simplify to get t^2 +4 -4t
      so that is where the -4t came from.
      (1 vote)
  • hopper jumping style avatar for user Rory Avera
    How do you know when to use the "inside out property" or the composing function?
    (8 votes)
    Default Khan Academy avatar avatar for user
  • starky ultimate style avatar for user Nigar Kainath
    (f ∘ g)(x)
    here, what does the sign ∘ mean?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • mr pants green style avatar for user awesomeness.RM
    Can someone please simplify all of this for me cause i am so confused!
    (2 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Jennifer Laessig
    If f(x)=(1/x) and (f/g)(x)=((x+4)/(x^2+2x)), what is the function g?
    (4 votes)
    Default Khan Academy avatar avatar for user
    • stelly blue style avatar for user Kim Seidel
      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.
      (2 votes)
  • female robot ada style avatar for user Jasmine Flores
    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?
    (3 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user Yvette Sanders
      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. (:
      (2 votes)
  • blobby green style avatar for user Arbaaz Ibrahim
    In the last question, is there a problem with the graph because, I'm finding it difficult to understand it?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • mr pants green style avatar for user ozbonus
      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
      (4 votes)
  • duskpin sapling style avatar for user bh
    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?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user Aditya Mahajan
    May someone please explain the challenge problem to me?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • mr pink orange style avatar for user Dylan Chan
      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
      (2 votes)
  • marcimus purple style avatar for user yulisa torres
    where can i find how to solve f(x)<g(x) using a graph?
    (1 vote)
    Default Khan Academy avatar avatar for user