Composing functions (article) | Khan Academy (2024)

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• Tess Van Horn

  8 years agoPosted 8 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

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  (66 votes)

  • Mr.Magroo

    8 years agoPosted 8 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.

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    (115 votes)
• Nigar Kainath

  8 years agoPosted 8 years ago. Direct link to Nigar Kainath's post “(f ∘ g)(x)here, what doe...”

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

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  (2 votes)

  • Levi Geadelmann

    8 years agoPosted 8 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)).

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    (15 votes)
• Rory Avera

  7 years agoPosted 7 years ago. Direct link to Rory Avera's post “How do you know when to u...”

  See Also

  Evaluate composite functions | College AlgebraGroup of Eight (G-8): Overview, History, CriticismsComposite Functions – Explanation & ExamplesHow to compose functions at a specific number

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

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  (9 votes)

  • Judith Gibson

    7 years agoPosted 7 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.)

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    (7 votes)
• Aditya Mahajan

  5 years agoPosted 5 years ago. Direct link to Aditya Mahajan's post “May someone please explai...”

  May someone please explain the challenge problem to me?

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  (2 votes)

  • Dylan Chan

    5 years agoPosted 5 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

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    (13 votes)
• flowermap21

  10 months agoPosted 10 months ago. Direct link to flowermap21's post “In question 4 how do peop...”

  In question 4 how do people get the 4t in tsquered-t4+9?

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  (3 votes)

  • Kim Seidel

    10 months agoPosted 10 months ago. Direct link to Kim Seidel's post “It comes from (t-2)^2(t-...”

    

    It comes from (t-2)^2
    (t-2)^2 = (t-2)(t-2) = t^2-2t-2t+4 = t^2-4t+4
    To square binomials, you need to use FOIL or the pattern for creating a perfect square trinomial. You can't square the 2 terms and get the right answer.

    Hope this helps.

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    (11 votes)

    See Also

    Composite Function Calculator
• magk2006

  10 months agoPosted 10 months ago. Direct link to magk2006's post “in the example question "...”

  in the example question "g(x)= x+4, h(x)= x(squared)-2x" how does it get the +8x and -2x in the distribute section ?
  here's the distribute equation =(x(squared)+8x+16−2x−8)
  ​

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  (5 votes)

  • Kim Seidel

    10 months agoPosted 10 months ago. Direct link to Kim Seidel's post “h(g(x)) = (x+4)^2 - 2(x+4...”

    h(g(x)) = (x+4)^2 - 2(x+4)
    Basically each "x" in h(x) gets replaced with (x+4), which if g(x). Then, you simplify.

    1) FOIL out (x+4)^2:
    h(g(x)) = x^2+4x+4x+16 - 2(x+4) = x^2 + 8x + 16 - 2(x+4)

    2) Distribute -2: h(g(x)) = x^2 + 8x + 16 - 2x - 8

    3) Combine like terms: x^2 + 6x + 8

    Hope this helps.

    Comment on Kim Seidel's post “h(g(x)) = (x+4)^2 - 2(x+4...”

    (6 votes)
• ScribofThoth

  10 months agoPosted 10 months ago. Direct link to ScribofThoth's post “I still can't get this. I...”

  I still can't get this. I think my problem is them showing multiple ways to do this instead of focusing on how to combine it into one equation. Either I have to work each function alone then combine them at the end or have more help figuring out how to make one equation.

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  (2 votes)

  • ersepsi

    10 months agoPosted 10 months ago. Direct link to ersepsi's post “I don't think their aim i...”

    I don't think their aim is to show you the multiple ways you can evaluate the composite function.

    The first example they basically show what evaluating a composite function really means, it's like you said "work each function alone". In the second example they showed a more faster and efficient way to evaluate the composite function by combining them into one equation.

    If you're still confused about composite functions, I'll explain this way:

    we have a function f(x), this function takes "x" as "input". Now, I'm certain you're used to the variable x being substituted for a number, but in maths, you can pretty much substitute it for anything you like. (Expressions for example)

    Like I can let x = 5, but I can also let x = 2h. Doesn't that mean I can also substitute x for some function? In other words x = g(x).

    Say if g(k) = 4k, then this would become: x = 4k. (Because x = g(k) = 4k)

    Since we let x = g(k) = 4k, then our function f can be written as: f( g(k) ) or f(5k) (We substituted x for g(k) )

    if f(x) = 5x, by substituting x for g(k), this becomes:

    f( g(x) ) = 5g(x) ---> f( 4k ) = 5(4k) = 20k

    This also means that our composite function changes value depending on the value of k.

    Conclusion: g(k) becomes input for function f.

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    (8 votes)
• awesomeness.RM

  8 years agoPosted 8 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!

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  (2 votes)

  • Kim Seidel

    8 years agoPosted 8 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

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    (6 votes)
• Jennifer Laessig

  7 years agoPosted 7 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?

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  (4 votes)

  • Kim Seidel

    7 years agoPosted 7 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.

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    (2 votes)
• x.asper

  8 months agoPosted 8 months ago. Direct link to x.asper's post “How do we know that g = 3...”

  How do we know that g = 3 in the first example study? I looked multiple times, and couldn't see where we found that value. Any help?

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  (3 votes)

  • Kim Seidel

    8 months agoPosted 8 months ago. Direct link to Kim Seidel's post “I assume you are asking a...”

    I assume you are asking about the first example on the page. The initial problem statement gives you the equations for f(x) and g(x). It then asks you to find f(g(3)).

    g(3) is part of what the problem is asking you to find. It doesn't say that g=3. It says uses the function g(x) with an input value of x=3.

    Hope this clarifies thing.

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    (3 votes)