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.
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...
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...
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...
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
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?
##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...
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.
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?
## 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...
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.]
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?
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)...
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...
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...
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...
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...
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
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...
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...
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
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...
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...
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...
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...
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?
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...
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??
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...
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...
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...
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...
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...
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 =...
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...
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...
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
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...
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?
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
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) +...
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...
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...
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...
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...
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...
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...
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.