What is Composite function: Definition and 73 Discussions

In mathematics, function composition is an operation that takes two functions f and g and produces a function h such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x in X to g(f(x)) in Z.
Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X.
The notation g ∘ f is read as "g circle f ", "g round f ", "g about f ", "g composed with f ", "g after f ", "g following f ", "g of f", "f then g", or "g on f ". Intuitively, composing functions is a chaining process in which the output of function f feeds the input of function g.
The composition of functions is a special case of the composition of relations, sometimes also denoted by






{\displaystyle \circ }
. As a result, all properties of composition of relations are true of composition of functions, though the composition of functions has some additional properties.
Composition of functions is different from multiplication of functions, and has quite different properties; in particular, composition of functions is not commutative.

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  1. C

    Finding domain for when composite function is continuous

    I am trying to find where ##h(x) =In{x^2}## is continuous on it's entire domain. My reasoning is since natural log is defined for ##x > 0##, then the argument ##x^2## should be positive, ##x^2 > 0##, we can see without solving this equation that ##x ≠ 0## for this equation to be true, however...
  2. C

    Using continuity to evaluate a limit of a composite function

    For this problem, The solution is, However, I tried to solve this problem using my Graphics Calculator instead of completing the square. I got the zeros of ##x^2 - 2x - 4## to be ##x_1 = 3.236## and ##x_2 = -1.236## Therefore ##x_1 ≥ 3.236## and ##x_2 ≥ -1.236## Since ##x_1 > x_2## then...
  3. C

    Finding the domain of a composite function

    For this problem, The solution is, However, I tried solving this problem by using the definition of composite function ##f(g(x)) = f(\frac{4}{3x -2}) = \frac{5}{\frac{4}{3x - 2} - 1} = \frac{5}{\frac{6 - 3x}{3x - 2}} = \frac {15x - 10}{6 - 3x}## which only gives a domain ##x ≠ 2##. Would some...
  4. S

    Solution of inequality of composite function involving inverse

    I can solve (i), I got x = -1.6 For (ii), I did like this: $$(f^{-1} o ~g)(x)<1$$ $$g(x)<f(1)$$ But it is wrong, the correct one should be ##g(x) > f(1)##. Why? Thanks
  5. greg_rack

    Solving an immediate indefinite integral of a composite function

    That's my attempt: $$\int (\frac{1}{cos^2x\cdot tan^3x})dx = \int (\frac{1}{cos^2x}\cdot tan^{-3}x) dx$$ Now, being ##\frac{1}{cos^2x}## the derivative of ##tanx##, the integral gets: $$-\frac{1}{2tan^2x}+c$$ But there is something wrong... what?
  6. S

    B Domain of a composite function

    ##f(x)=-x^2 + 3## so ##f^{-1} (x)=- \sqrt{3-x} ~, x \leq 3## ##ff^{-1} = - (- \sqrt{3-x})^2 + 3 = x## and the domain will be ##x \leq 3## ##f^{-1} f = - \sqrt{3-(-x^2+3)} = -x ## and the domain will be ##x \leq 0## My question: ##ff^{-1} (x)## or ##f^{-1} f(x) ## is not always equal to...
  7. rxh140630

    Formulas for computing composite function

    h(x) = 0 for x ≤ 0 h(x) = x^2 for x>0 But my book says h(x) = 0 for x<0 h(x) = x^2 for x≥0 Can my solution (the first one) work as well? Because the actual function value at x = 0 is zero. I feel like my solution is more elegant.
  8. anemone

    MHB Solving 8 Roots: Can 3 Quadratic Polynomials Fulfill $f(g(h(x)))=0$?

    Is it possible to find three quadratic polynomials $f(x),\,g(x)$ and $h(x)$ such that the equation $f(g(h(x)))=0$ has the eight roots 1, 2, 3, 4, 5, 6, 7 and 8?
  9. baldbrain

    Range of f(x): Intersection of h(x) and g(x) Ranges

    ## Let~~f(x)=h(x)+g(x) , where~~h(x)=10^{\sin x}~~and~~g(x)=10^{\csc x}## ##Then,~~D_f = {D_h}\cap {D_g}## ##Clearly,~~D_h=ℝ~~and~~D_g=ℝ-\{nπ|n∈ℤ\}## ##∴~~D_f =ℝ-\{nπ|n∈ℤ\}## After considering the new domain, the range of ##\sin x## in ##10^{\sin x}## is ##[-1,1]-\{0\}## Therefore, the range of...
  10. L

    How does this composite function simplify to 2(2^x) ?

    f(x)=2xand g(x)=2^x Find the composite function of fg(x) fg(x) =f(g(x)) =f(2^x) =2(2^x) I don’t understand how this in turn equals to 2^(x+1) [Moderator's note: Moved from a technical forum and thus no template.]
  11. Akash47

    Finding f(6) from a composite function

    It is obvious that the function f is not injective. From the given equation, we get f(f(2))=6.And since,there is an inequality given in the problem, I think we can use that to find f(6).But I have got stuck here and can't move.Do I have to find what is f(x) first?Then how?
  12. E

    Curiosity about why this is not a Function composition

    Homework Statement If ##f\left(x\right) = x^2 + 2x + 2##, find two functions ##g## for which ##\left(f \circ g\right)\left(x\right) = x^2 - 4x + 5##. Homework Equations If ##f\left(x\right) = x^2 + 2x + 2##, then ##\left(f \circ g\right)\left(x\right) = g\left(x\right)^2 + 2g\left(x\right) +...
  13. N

    Find the Composite Function of p & q: Relationship & Value of pq(39.72)

    Homework Statement p(x)=(2−x) / (3 + x) and q(x)= (2−3x) / (1+ x) a) Find the function pq(x). b) Hence describe the relationship between the functions p and q. c) Hence write down the exact value of pq(39.72). 2. The attempt at a solution a) I got pq(x) = x by substituting q(x) into p(x)...
  14. N

    Solving Composite Functions: Understanding the Addition Rule for Division

    Homework Statement and Find , and 2. The attempt at a solution means work out , then work out for this value. so I do not understand why we add + 3 in line before last (2 x 16 + 3) Should it not be divided by 3 as it is which means 3 x g(x) = x^2 I am a little stuck on the...
  15. D

    B Domain and the codomain of a composite function

    So, I'm a bit confused. The thing is, basically, all elementary functions are of the form ƒ:ℝ→ℝ. So the domain is ℝ and so is the codomain. However, if we have a function ƒ:ℝ→ℝ, given with f(x) = √x, it's domain is now x≥0. So, is the domain of this function ℝ or [0,+∞>? Also, let's say we have...
  16. lawsonfurther

    A A bit of clarification on the domain of a composite function

    Recently when I reviewed something about the composite function for my calculus exam, I remembered I had been thinking a question for quite a long time (maybe I was going into a dead end) since I was in high school. I was thinking whether f(x)=1/(1/x) and g(x)=x are the same function or not. It...
  17. S

    A How to find the partial derivatives of a composite function

    Hello, dear colleague. Now I'm dealing with issues of modeling processes of heat and mass transfer in frozen and thawed soils. I am solving this problems numerically using the finite volume method (do not confuse this method with the finite element method). I found your article: "Numerical...
  18. S

    Unknown composite function

    Homework Statement Find f(x) given that f ( f(x) - x2) = x2 - 5x + 3 Homework Equations Not sure The Attempt at a Solution I tried assuming f(x) = ax + b and use composite function but end up wrong. Please give me idea to start Thanks
  19. Quadrat

    Range of composite function

    Homework Statement [/B] The function ##f##, ##{f: ℤ → ℚ}## defined as ##f(a)=cos(πa)## The function ##g##, ##{g: ℚ→ ℝ}## defined as ##g(a)=(5a)/4## Let h be the composite funciton ##h(a)=f(g(a))## What's the range of this function h? Homework Equations [/B] ##h(a)=cos(5πa/4)## The domain...
  20. toforfiltum

    Conflicting result in derivative of composite function

    Homework Statement Let $$f(x,y)=\begin{cases} \frac{x^2y}{x^2+y^2} \space & \text{if} \space(x,y)\neq(0,0)\\0 \space & \text{if} \space(x,y)=(0,0)\end{cases}$$ a) Use the definition of the partial derivative to find ##f_x(0,0)## and ##f_y(0,0)##. b) Let a be a nonzero constant and let...
  21. 5

    Find derivative of composite function

    Homework Statement [/B] Consider the equation z=6x8ln(x) where z and x are functions of t.If dx/dt=5 when x=e calculate dz/dt. Homework Equations [/B] Do I have to rearrange the equation to do this?The Attempt at a Solution
  22. S

    Limit of Composite Function

    I need help with the following theorem: Let I, J ⊆ℝ be open intervals, let x∈I, let g: I\{x}→ℝ and f: J→ℝ be functions with g[I\{x}]⊆J and Limz→xg(x)=L∈J. Assume that limy→L f(y) exists and that g[I\{x}]⊆J\{g(x)},or, in case g(x)∈g[I\{x}] that limy→L f(y)=f(L). Then f(g(x)) converges at x, and...
  23. P

    Longtime lurker here, composite function problem

    Homework Statement Let f(g(h(x))) = 1/(2-x) Find g(x) if: f(x) = (x^2) - 1 h(x) = 3x+12. The attempt at a solution This is what I have: g(h(x))^2 -1 = 1/(2-x) g(h(x) = sqrt((3-x)/(2-x)) I'm not sure how to get the h(x) out of this to leave me with just g(x). Please point me in the right...
  24. N

    Confused about composite functions

    Homework Statement 1. Find a formula for (f g)(x) = ? 2. Find a formula for (f f )(x) = ? 3. Find a formula for the composition below. g(h(x)) = 4. Find a formula for the composition below. (h g)(x) =The Attempt at a Solution 1. f(g(x)) 2. f(f(x)) 3. (g º h)(x) 4. h(g(x)) Why are these...
  25. I

    Domain of a composite function

    Homework Statement Given the Functions f(x)=4x-1 g(x)=3-2x^2 h(x)= sqrt (x+5) What is the domain of h(g(x))? Homework Equations the subject is finding the domain of a composite function The Attempt at a Solution I don't understand what I have to 'bring over' from g(x). I think x cannot equal...
  26. J

    Functional and composite function

    What is the difference between a functional and a composite function? Also, look those implicit equations: ##F(x, y(x))=0##, ##F(t, \vec{r}(t))=0##, ##F(x, y(x), y'(x), y''(x))=0##, ##F(t, \vec{r}(t), \vec{r}'(t))=0##... Can be understood that ##F## is the functional?
  27. L

    Composite function of a piecewise function

    Homework Statement Given that I have a doubling function : f(x)=2x (for 0≤x<0.5) and 2x-1 (for 0.5≤ x<1) Homework Equations What is f(f(x))? The Attempt at a Solution f(f(x))=4x for the first one and 4x-3 for the second part but not sure what to do about the domain...
  28. C

    Domain of composite function

    Let's say f(x) = sqrt(x-1) And g(x) =x^2 What is the domain of f(g(x)) Well the domain of g(x) is all real numbers and the equation for the new function is sqrt(x^2-1) Am I right to say that the domain for f(g(x)) is x greater than or equal to 1 and less than and equal to -1??
  29. C

    Inputs and outputs of composite function

    Homework Statement Can someone describe the input and output method to find the domain and ranges of composite functions to me?? Homework Equations The Attempt at a Solution Example: f(g(x)) first you find the domain of g(x) Then you find the range of g(x) y= G(x) that will...
  30. T

    Integration of a composite function

    y=f(x),K=F(y), and dF(y)/dy=y^2/y', (1) then dF(x)=y^2*dx; so, F(x)=int(y^2*dx)=int((f(x)^2)*dx); then we obtain, F(x)=(f(x))^3/(3*f'(x))+C; substitution of y=f(x) into F(x), we get, F(y)=y^3/(3*y')+C; (2) using the result...
  31. T

    Derivative of a composite function

    y =f(x), and y'=df(x)/dx, F(y)=y^3+(y')^2 how to deal with the (y')^2 when i calculate dF(y)/dy? thanks.
  32. M

    MHB Composite function troubles

    g(x)=−4x2−5x f(x)=−3x2+7x−5(g(x)) f(g(−1))=? First, let's solve for the value of the inner function, g(−1). Then we'll know what to plug into the outer function. g(−1)=−4(−1)2+(−5)(−1)I don't understand why they transformed the minus symbol into an addition symbol. This has happened a few...
  33. Petrus

    MHB Integrate Sine and Square root Composite Function

    Hello MHB, I got stuck on integrate this function \int \frac{\sin^3(\sqrt{x})}{\sqrt{x}}dx my first thinking was rewrite it as \int \frac{\sin^2(\sqrt{x})\sin(\sqrt{x})}{\sqrt{x}}dx then use the identity \cos^2(x)+\sin^2(x)=1 \ \therefore \sin^2x=1- \cos^2(x) \int...
  34. B

    One to one and onto in composite function

    Homework Statement I just want to make sure that I am correct. if we have a composite function f(g(x)). Homework Equations f(g(x)) is onto if and only if both f(x) and g(x) are onto f(g(x)) is one to one if and only if or both f(x) and g(x) are one to one The Attempt at a Solution...
  35. I

    Invert a triple composite function p(q(r(x)))

    Hey, Let ##(f,g) \in B^A## where ##A## and ##B## are non-empty sets, ##B^A## denotes the set of bijective functions between ##A## and ##B##. We assume that there exists ##h_0: A \rightarrow A## and ##h_1: B \rightarrow B## such that ##f = h_1 \circ g \circ h_0 ##. This implies that ##g =...
  36. Saitama

    Composite function and definite integration

    Homework Statement f(x)=x^3-\frac{3x^2}{2}+x+\frac{1}{4} find\int_{\frac{1}{4}}^{\frac{3}{4}} f(f(x))dx Homework Equations The Attempt at a Solution I am clueless here. I started by writing f(f(x)) as (f(x))^3-\frac{3(f(x))^2}{2}+f(x)+\frac{1}{4} I don't think expanding...
  37. A

    Help with Composite Function Derivatives

    1. If F(x) = f(xf(xf(x))), where f(1) = 2, f(2) = 3, f '(1) = 4, f '(2) = 5, and f '(3) = 6, find F'(1). I feel I have a decent grasp on the chain rule, product rule, etc, but when faced with a problem like this I just blank out. I don't even really know where to begin. Unfortunately I...
  38. A

    Second derivative composite function

    Hi guys, I have this function f(g(t)) and I have to find the second time derivative of f, is it correct the following solution?: f''=∂f/∂g*g'=∇f*g' f ''=∇^2f*|g'|^2+∇f*g'' where ∇^2 is the laplacian function
  39. nukeman

    Finding the domain of a composite function help.

    Homework Statement Ok, I just worked out a composite function, and it left me with: √2x^2+5) Now, how do I find the domain from that? I don't understand that my text says the domain of that is just all Real numbers ? What makes this different than other square functions that we are...
  40. nukeman

    Composite function help ? - Thanks

    Homework Statement Here is the problem: Homework Equations The Attempt at a Solution I need help RIGHT from step one. Now, step one I would suppose I need to evaluate g(x) ? Which, would be x must be greater than or equal to -5. Correct? Then what?
  41. N

    Integration of a composite function

    Homework Statement The question I have is a more general one, but one I can't find an anser to since I don't have any access to a book on integration at the moment. How do we Integrate a composite function. ∫f(g(x)) dx Homework Equations The Attempt at a Solution
  42. J

    Composite Function with reccursive expression

    Hi guys I'm thinking on this problem for long time :) Let be one function defined only on POSITIVE INTEGERS I) f(1) = 1 II) f(2n) = 2 . f(n) + 1, if n ≥ 1 III) f(f(n)) = 4n + 1, if n ≥ 2 Find f(1993) ______________ First I got this, I got to this milestone f(2n) = 2 . f(n) +...
  43. V

    Continuation of composite function problem problematic

    I asked about the first part of this problem in https://www.physicsforums.com/showthread.php?t=592408. I thought the best idea was to start another thread for the second part. Homework Statement Part a) Given f(x)=ax+b, and f(3)(x)=64x+21, find the values of the constants a and b. (note...
  44. V

    Composite function problem.

    Given f(x)=ax+b, and f(3)(x)=64x+21, find the values of the constants a and b. (note: f(3)(x) means fff(x)) To me this seems like I have to use two equations to find the value of three variables, since when I have found a and b, I should be able to get the value of x. Even though it...
  45. G

    Domain and range of composite function

    Homework Statement The function f has domain (-∞, ∞) and is defined by f(x) = 3e2x. The function g has domain (0, ∞) and is defined by g(x) = ln 4x. (a) Write down the domain and range of f∘g. (b) Solve the equation (f∘g)(x) = 12 2. The attempt at a solution (a) Is it correct...
  46. J

    Right limit of a composite function. Original functions limits are known.

    Homework Statement If \lim_{x\rightarrow 0+}f(x)=A and \lim_{x\rightarrow 0-}f(x)=B find \lim_{x\rightarrow 0+}f(x^{3}-x) Homework Equations The Attempt at a Solution I don't have one. I am dumbfounded. Mostly i have been trying to understand the meaning of the composite...
  47. K

    Derivative of Composite Function

    Homework Statement \begin{equation} f(x)= \begin{cases} 5x+2 &, x \leq 1 \\ 3x^2 &, 1<x<2\\ 4-x &, x\geq 2 \end{cases} \end{equation} \begin{equation} g(x)= \begin{cases} \frac{1}{5}(2+3 cos x) &, x <0 \\ 4-sin x &, x \geq 0 \end{cases} \end{equation} Find h = f...
  48. T

    Proving the graph of a composite function

    Homework Statement Let f : A → B, g : B → C be functions where A,B,C are sets. ConsiderΓf ⊂A×B,the graph of f,Γg ⊂B×C,the graph of g. Now consider the sets Γ f ×C ⊂ A×B×C and A×Γg ⊂A×B×C. LetΓ=θ(Γf ×C∩A×Γg)⊂A×C where θ : A×B×C → A×C is the projection defined as θ((a,b,c))=(a,c). Show that Γ...
  49. L

    Finding the domain of a composite function

    Homework Statement I started out with f(x)=sinx and g(x)=1-√x. I found f(g(x)) which is sin(1-√x) and now my problem is how to find the domain. I've really been struggling with the domain part and just need this one done step by step so i have an idea of how to actually do it. Homework...
  50. P

    Range of a Composite Function

    Function p and q are defined by: p (x) = 3x2+1, x∈R, Domain 0≤x≤2 q (x) = x2 - 2, x∈R (q∘p) - State the range: I got -1 ≤ y ≤ 167 The book says 0 ≤ y ≤ 167 Any idea where I went wrong? The composite function I got so then I could sub was: 9x4 + 6x2 - 1 Thanks, Peter G.
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