What is Composite function: Definition and 74 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. P

    Why Does My Range for a Composite Function Differ from the Textbook's?

    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.
  2. N

    Integrating Exp, Trig Composite function

    Homework Statement Hey, I've been working through a book and one problem just gets me that I know should be a piece of cake. I don't know if I'm just being an idiot or not seeing something but the problem is to take int e^(ax)cos(bx)dx and int e^(ax)sin(bx)dx simultaneously by multiplying the...
  3. G

    What is the Limit of a Composite Function?

    Homework Statement If lim f(x) as x->0 is = 0 then lim \frac{sin(f(x))}{f(x)} as x->0 = 1? dont know how to start proving this . thanks for the replies
  4. D

    Is (h\circ g)\circ f = h\circ (g\circ f)?

    1. Prove that (h\circ g)\circ f = h\circ (g\circ f) Homework Equations f:A\longmapsto B, g:B\longmapsto C, h:C\longmapsto D The Attempt at a Solution (h\circ g)\circ f =\{(b,d):d=h(c)\}\circ f =\{(b,d):d=h(g(b))\}\circ f I reach there and get stuck to continue...
  5. J

    Finding Discontinuous Composite Function f o g at x = 0

    Homework Statement I need to find functions f and g both continuouis at x = 0 for which the composite f at g is discontinuous at x = 0 Homework Equations The Attempt at a Solution I thnk it is a matter of looking for a composite function that results in 0 being in the...
  6. S

    Prove a composite function is increasing

    Homework Statement Hi, I have trouble proving this claim and would really appreciate your help =). Thank you in advance! So here's the question: Suppose that f is a continuous function for all x>= 0 and differentiable for all x> 0. Also, f(0) = 0 and f' (1st derivative of f) is increasing on...
  7. estro

    Uniform continuity of composite function

    I'll be very thankful is someone will tell me where I'm wrong. We know: 1) f is uniform continuous. 2) g is uniform continuous. We want to prove: fg(x) is uniform continuous. proof: from 1 we know -> for every |a-b|<d_0 exists |f(a)-f(b)|<e from 2 we know -> for every |x-y|<d...
  8. T

    What is the range of the composite function gf(A)=C?

    Homework Statement The sets A and B are defined respectively by A={x\in R : 0\leq x\leq 1} B={x\in R : 1\leq x\leq 2} and the functions f and g are defined respectively by f(x)=x^2-2x+2 g(x)=\frac{x+2}{x-1} where f(A)=B , g(B)=C with C as the range of the function g ...
  9. I

    Proof of Theorem: Composite Function Inverse

    i really need to see the proof of this theorem: if f and g are bijective then the inverse of (g o f) = inverse of f o inverse of g
  10. N

    Composite Function Homework: Proving One-to-One & Onto

    Homework Statement let A,B,C be sets, and let f : A--> B and g : B--> C be functions. The composite function denoted by g o f is a function from A to C defined as follows: Homework Equations g o f(x)=g(f(x)) for every x in A. Prove that if g o f is one-to one, then f is one-to...
  11. C

    How do you determine equation for this simple composite function?

    Homework Statement Given g(x)=|x| and f(x)=secx 1)Determine an equation for (f(g(x)) 2)State the domain 3)x and y intercepts 4)any asymptotes Homework Equations N/A The Attempt at a Solution f(g(x))=sec|x| or 1/cos|x|
  12. L

    Implicit Differentiation w/ Composite Function

    Homework Statement Given the equation y= f(x) , at a certain point the slope of the curve is 1/2 and the x-coordinate decreases at 3 units/s. At that point, how fast is the y-coordinate of the object changing? The Attempt at a Solution Dy/dx = f ' (x) dx/dt Would that be...
  13. P

    Composite Function Homework: Is My Solution Correct?

    Homework Statement The attempt at a solution g(f(x)) = h(x) 4f(x) + y = 4x - 1 4x + 16 + y = 4x - 1 y = -1 - 16 y = -17 so, g(x)= 4x + y = 4x - 17 Is this the correct way of going about this question? I used a guessing approach to this question. Is enough work shown to get...
  14. L

    Solve Composite Function: g(f(x)) = 225?

    Homework Statement If f(x) = x3+3x2+4x+5 and g(x)=5, then g(f(x)) = ? Homework Equations The Attempt at a Solution I don't know if I am correct. g(f(x)) = g(x3+3x2+4x+5)= 225? I plugged 5 into the equation. Am I right?
  15. P

    Solve Composite Function Homework: f(g(h(x)))

    Homework Statement http://img171.imageshack.us/img171/543/pro3.png The Attempt at a Solution = f(g(h(x))) = f(g(\sqrt{x+3})) = f(cos\sqrt{x+3}) = \frac{2}{cos\sqrt{x+3}+1} Is this right, and am I showing enough work to get full marks? Do I need restrictions for example?
  16. T

    Is the Gradient of a Composite Function Always Zero?

    Homework Statement http://img21.imageshack.us/img21/8175/46521897.jpg The Attempt at a Solution I think I have a starting point, but I'm not 100% sure Basically I thought of just computing grad(f(α(t)) · dα/dt and showing its equal to zero. Am I on the right track, or shall I try...
  17. P

    Derivative of a composite function?

    Homework Statement Find the derivative of the function: cos(x)^(cos(cos(x))) Homework Equations The chain rule The Attempt at a Solution I know how the chain rule works, and I've done many problems with composite functions. However, I just don't know where to start with this one...
  18. L

    Discuss continuity of the composite function

    Homework Statement : discuss continuity of the composite function h(x)=f(g(x)) when A} F(x)=X^2 , g(x) = x-1 B} f(x) = 1/x-6 , g(x) = X^2+5 where should I start ?
  19. I

    Is this correct?Is f onto if g∘f is onto and g is one-to-one?

    Homework Statement Let f: X\rightarrowY and g: Y\rightarrowZ be functions. Prove or disprove the following: if g\circf is onto and g is one-to-one then f is onto. Homework Equations N/A The Attempt at a Solution I'm honestly not sure what to do with this. I believe that the...
  20. H

    Integration of an inverse sqrt composite function

    Homework Statement “Geologist A” at the bottom of a cave signals to his colleague “Geologist B” at the surface by pushing a 11.0 kg box of samples from side to side. This causes a transverse wave to propagate up the 77.0 m rope. The total mass of the rope is 14.0 kg. Take g = 9.8 m/s². How...
  21. M

    Composite function and continuerty

    Hello I have the following problem: Given two function f and g which are continuer on R, and some point c which belongs to R. I'm suppose to show that if f(c) = g(c), then h is continious on R. Isn't that the same as showing that that the composite function h(c) = g(f(c)) is...
  22. N

    Mastering the Chain Rule for Complex Derivatives

    I understand hhow to use the chain rule for a simple 2 part composite function, however, I tend to have problems when it gets past that. Can someone please help me master these complex derivatives, or just a few quick tips would be nice --Thanks
  23. X

    Solving for f in f(f(x)) = 2x^2 - 1

    What is the f, when f(f(x)) = 2x^2 - 1, Thanks alot! :confused:
  24. F

    The domain of a composite function

    my previous calculus teacher stressed finding the domain of a composite function; he stressed more of finding which areas were not part of the function by means of three circles and saying the function didn't pass to the other circle unless it met the range of the other function. The textbook...
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